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Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence HW1 – Grade C Name: Sequences 1. (a) The first term of a sequence is –2. The rule for continuing the sequence is Teacher Assessment Add 7 then Multiply by 4 What is the second term of the sequence? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………………… (1) (b) This rule is used to continue a different sequence. Multiply by 2 then Add 5 The second term of this sequence is 3. What is the first term? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………………… (3) (Total 4 marks) The Robert Smyth School 1 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 2. A pattern using pentagons is made of sticks. (a) Diagram 1 Diagram 2 Diagram 3 5 sticks 9 sticks 13 sticks How many sticks are needed for Diagram 5? ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (2) (b) Write down an expression for the number of sticks in Diagram n. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (2) 3. Patterns are made from shaded and unshaded squares. 1st pattern (a) 2nd pattern 3rd pattern 4th pattern How many shaded squares are there in the nth pattern? …………..........…............……...................…………….....…………...................... Answer ………………………………….. (1) (b) How many unshaded squares are there in the nth pattern? …………..........…............……...................…………….....…………...................... …………..........…............……...................…………….....…………...................... Answer ………………………………….. (1) (Total 2 marks) The Robert Smyth School 2 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 4. The diagrams show a sequence of rectangles. Area = 2 Diagram 1 (a) Area = 6 Diagram 2 Area = 12 Diagram 3 Area = 20 Diagram 4 Write down an expression for the area of Diagram n. ..................................................................................................................................... ..................................................................................................................................... Answer ...................................................................... (2) (b) A rectangle in the sequence has an area of 110 cm2. What is its perimeter? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ................................................................ cm (2) (c) Which diagram has a perimeter of 26cm? ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. ...............…………………………………………………………………………….. Answer ............................................ (3) (Total 7 marks) 5. (a) The nth term of a sequence is 4n + 1 (i) Write down the first three terms of the sequence. .......................................................................................................................... .......................................................................................................................... .......................................................................................................................... Answer ......................., ......................, ....................... (2) The Robert Smyth School 3 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence (ii) Is 122 a term in this sequence? Explain your answer. .......................................................................................................................... .......................................................................................................................... (1) (b) Tom builds fencing from pieces of wood as shown below. Diagram 1 4 pieces of wood Diagram 2 7 pieces of wood Diagram 3 10 pieces of wood How many of pieces of wood will be in Diagram n? ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer ....................................................... (2) (Total 5 marks) 6. (a) Write down the first three terms of the sequence whose nth term is given by 5n 4n 7 ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (b) Which term of the sequence has a value of 1? ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer .................................................... (2) (Total 4 marks) Success: The Robert Smyth School Target: 4 The Robert Smyth School Mathematics Faculty HW2 – Grade C Topic 5 – Expressions, Formulae and Sequences Innovation & excellence Sequences 1. Teacher Assessment The nth term of a sequence is given by the expression n2– 3 Write down the first three terms of the sequence. ............................................................................................................................................... ............................................................................................................................................... Answer .................... , ..................... , ...................... (Total 2 marks) 2. (a) Write down the first three terms of the sequence whose nth term is given by 5n 3. ..................................................................................................................................... ..................................................................................................................................... Answer........................................................................................................................ (2) (b) The letters p and q represent integers. p is an odd number and q is an even number. (i) Which of these statements describes the number 2p + q? always even always odd could be odd or even Explain your answer. ........................................................................................................................... ........................................................................................................................... (2) (ii) 1 q is always even. 2 Give an example to show that Anna is wrong. Anna says that p + ........................................................................................................................... ........................................................................................................................... (1) (Total 5 marks) The Robert Smyth School 5 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 3. (a) p is an odd number. Is 2p + 1 an odd number, an even number or could it be either? Tick the correct box. odd even either (1) (b) The nth term of a sequence is 4n + 1 (i) Write down the first three terms of the sequence. ........................................................................................................................... ........................................................................................................................... Answer .......................................................... (2) (ii) Is 122 a term in this sequence? You must explain your answer. ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... (1) (Total 4 marks) 4. A sequence of numbers is shown. 2 (a) 5 8 11 14 Find an expression for the nth term of the sequence. ...................................................................................................................................... ...................................................................................................................................... Answer ......................................................................... (2) (b) Explain why 99 will not be a term in this sequence. ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... (2) (Total 4 marks) The Robert Smyth School 6 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 5. A sequence of rectangular patterns is shown. Pattern 1 (a) Pattern 2 Pattern 3 Pattern 4 Pattern 5 Calculate the number of small squares in Pattern 20. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ......................................................................... (2) (b) Explain why the number of small squares in Pattern n is n(n + 2). ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (2) (Total 4 marks) 6. Here is a number sequence. 1 (a) 3 6 10 15 Write down the next two numbers in the sequence. ...................................................................................................................................... Answer …………………………………......... (2) (b) Describe a rule for continuing the sequence. ...................................................................................................................................... ...................................................................................................................................... (1) The Robert Smyth School 7 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence (c) (i) x2 + x Work out the value of when x = 5 ............................................................................................................................ Answer …………………………………......... (1) (ii) Factorise x2 + x ............................................................................................................................ Answer …………………………………......... (1) Darren says that x2 + x is always even. (iii) Using your Answer to part (ii), or otherwise, explain why this is true. ............................................................................................................................ ............................................................................................................................ ............................................................................................................................ (2) (Total 7 marks) 7. Part of a number grid is shown below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 The shaded shape is called L3 because it has the number 3 at the top. The sum of the numbers in L3 is 26. (a) Calculate the sum of the numbers in L22 ..................................................................................................................................... Answer …………………………………......... (1) The Robert Smyth School 8 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence (b) Fill in the empty squares of Ln n (2) (c) Write down an expression, in terms of n, for the sum of the numbers in Ln Simplify your expression. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………......... (2) (d) If the sum of the numbers in Ln is 143, find the value of n. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …………………………………......... (2) (Total 7 marks) Success: The Robert Smyth School Target: 9 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty HW3 – Grade A Innovation & excellence Sequences 1. Teacher Assessment Find the nth term of the following sequences. (a) 1, 4, 9, 16, 25, ... ..................................................................................................................................... Answer............................................................................... (1) (b) –2, 1, 6, 13, 22, ... ..................................................................................................................................... ..................................................................................................................................... Answer............................................................................... (1) (Total 2 marks) 2. A sequence of numbers is shown. 5 (a) 9 13 17 21 Find an expression for the nth term of the sequence. ...................................................................................................................................... ...................................................................................................................................... Answer ................................................................ (2) (b) Explain why 83 will not be a term in this sequence. ...................................................................................................................................... ...................................................................................................................................... (2) (Total 4 marks) The Robert Smyth School 10 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 3. The triangle number sequence is 1, 3, 6, 10, 15, 21, .......... The nth term of this sequence is given by 1 n(n + 1) 2 (a) Write down an algebraic expression for the (n – l)th term of the sequence. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ....................................................... (1) (b) Prove algebraically that the sum of any two consecutive triangle numbers is a square number. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (3) (Total 4 marks) The Robert Smyth School 11 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 4. (a) Part of a number grid is shown below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 The shaded cross is called C11 because it has the number 11 at the centre. This is Cn n–7 n Fill in the empty boxes. (2) (b) Kevin notices the following number sequence in the grid. 1, 9, 17, 25, 33,... Write down the nth term of this sequence. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer ....................................................... (2) (Total 4 marks) Success: The Robert Smyth School Target: 12 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics HW4 – GradeFaculty B-A Innovation & excellence Rearranging Formulae 1. Make x the subject of the formula 3x + 2y = 8y – 3 Teacher Assessment Simplify your answer as much as possible. …........................................................................................................................................... …........................................................................................................................................... …........................................................................................................................................... …........................................................................................................................................... …........................................................................................................................................... …........................................................................................................................................... Answer x = .......................................................... (Total 3 marks) 2. Make t the subject of the formula w= t –v ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer t = ……………………………………….. (Total 2 marks) 3. Make x the subject of the formula w = x2 + y ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer x = ............................................................. (Total 2 marks) The Robert Smyth School 13 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 4. u t 5 3 Make t the subject of the formula .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. Answer t = ............................................ (Total 2 marks) 5. 1 s (u v) t 2 Make u the subject of the formula ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer u = ............................................................... (Total 3 marks) 6. Rearrange y xy 2 3x 4 to make x the subject. Simplify your answer as much as possible. ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer ....................................................................... (Total 4 marks) The Robert Smyth School 14 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 7. Make r the subject of the formula r – 3 = (t – 2r) ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ Answer ........................................................ (Total 4 marks) 8. (a) Make x the subject of x2 + k = 16 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer x = ......................................................... (2) (b) Make P the subject of A=P+ PRT 100 ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Answer P = ......................................................... (3) (Total 5 marks) The Robert Smyth School 15 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 9. Make x the subject of the formula y 3x 4 x–3 .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. .........................…………………………………………………………………………….. Answer x = ....................................... (Total 4 marks) 10. Make x the subject of the formula a(x – b) = a2 + bx ……….......................…............……...................…………….....…………....................... ……….......................…............……...................…………….....…………....................... ……….......................…............……...................…………….....…………....................... ……….......................…............……...................…………….....…………....................... ……….......................…............……...................…………….....…………....................... Answer ………………….…………….. (Total 4 marks) The Robert Smyth School 16 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 11. (a) Make c the subject of the formula E = mc2 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... Answer …................................................................... (2) (b) Make m the subject of the formula 1 E mgh mv 2 2 ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... (2) (Total 4 marks) The Robert Smyth School 17 Topic 5 – Expressions, Formulae and Sequences The Robert Smyth School Mathematics Faculty Innovation & excellence 12. A shape is made from two trapezia. a b a h h Not drawn accurately The area of this shape is given by A = h (a + b) + b (a + h) 2 2 Rearrange the formula to make a the subject. ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... Answer a = ………………………………….. (Total 4 marks) Success: The Robert Smyth School Target: 18
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