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TOPIC#08 Problem Solving Q1. Ans. Part of a pattern of numbers is shown in the table below. 1 2 3 4 5 n 5 8 11 14 p x 4 9 16 25 q y 1 8 27 64 r z 10 25 54 103 s t (a) Study the patterns and write down the value of p, the value of r and the value of s. (b) Find expressions, in terms of n, for each of x, y, z and t. (a) p = 14 + 3 = 17 q = (5 + 1)2 = 62 = 36 r = 53 = 125 s = 17 + 36 + 125 = 178 (b) x =2 + 3n y = (n + 1)2 z = n3 t = n3 + n2 + 5n + 3 Q2. 10. Consider the sequence 13 -2, 23 -4, 33 -6, (a) Write down the 5th term of the sequence. (b) Write down, in terms of n, and expression for the nth term of the sequence. (c) Evaluate the 10th term of the sequence. Ans. (a) 115 Q3. Find Ans. (b) 43 -8, … . n3 – 2n (c) 980 (a) the missing number in the sequence, 1, 3, 6 …., 15, 21, 28, (b) the 7th term in the sequence whose nth term is 3n – 1, (c) an expression, in terms of n, for the nth term of the sequence 5, 9, 13, 17, 21, …… (a) 10 (b) 20 (c) 4n + 1 1 TOPIC#08 Problem Solving Q4. It is required to find a rational number which is equal to the recurring decimal number 0.737373 … . (a) Given that x = 0.737373 …, find the value of 100x – x. (b) Hence express 0.737373 … in the form Ans. (a) 73 Q5. The natural numbers 1, 2, 3, … are written, in a clockwise (b) a , where a and b are integers. b 73 99 direction, on a circular grid as shown in the diagram. There are four numbers in each ring. The number 1, 2, 3, and 4 are in the first ring. The numbers 5, 6, 7 and 8 are in the second ring. The following numbers fill up the other ring in the same way. (a) Write down the numbers in the fourth ring. (b) Write down the largest number in the tenth right. (c) The sum, Sn, of the four numbers in the nth ring, where n = 1, 2 and 3, is given in the table below. Ans. n 1 2 3 Sn 10 26 42 4 (i) Write down the value of S4. (ii) Find, in its simplest form, an expression, in terms of r, for Sr. (iii) In which ring is the sum of the four numbers equal to 1018? (a) 13, 14, 15, 16 (c) (i) S4 = 58 (b) 40 (ii) Sr = 16r -6 (iii) In the 64th ring. 2 TOPIC#08 Problem Solving Q6. Look at this pattern (a) (b) Ans. (a) (b) Q7. 12 – 02 = 1 22 – 12 = 3 32 – 22 = 5 42 – 32 = 7 . . . . . . . . . Write down (i) the 8th line of the pattern, (ii) the nth line of the pattern. Use the pattern to find (i) 3402 – 3392, (ii) the integers x and y such that x2 – y2 = 701. (i) 82 – 72 = 15 (ii) n2 – (n – 1)2 = 2n – 1 (i) 679 x = 351; y = 350 (ii) Look at this pattern. 12 – 02 = 1 22 – 12 = 3 32 – 22 = 5 42 – 32 = 7 (a) (b) . . . . . . . . . Write down (i) the 7th line of the pattern, (ii) the nth line of the pattern. Use the patter to find (i) 2502 - 2492, 3 TOPIC#08 Problem Solving Ans. Q8. (ii) the integers x and y such that x2 – y2 = 801. (a) (i) 13 (ii) 2n - 1 (b) (i) 499 (ii) x = 401, y = 400 Bob makes fences using identical metal rods one metre long. The rods are bolted together at their ends. Some fences, With different lengths, Are shown below. (a) Write down the values of p and q. (b) Given that B = 3n + k, where k is a constant, find the value of k. (c) Find an expression for R in terms of n. (d) Bob has 200 bolts and 400 rods. How many complete fences can he make which have a length of 6 m? Ans. (a) p = 14 (d) a fence of length 6m needs, bolts = 20, and rods = 41. , q = 27 (b) k=2 (c) R = 7n - 1 400 = 9.756, 41 Bob can make 9 complete fences. Q9. 1 2 1 2 (a) Write down the next two terms in the sequence 20,16 ,13,9 , 6,.......... (b) Write down an expression, in terms of n, for the nth term of the sequence 4 TOPIC#08 Problem Solving 1, 4, 7, 10, 13, …………… 1 2 , 1 2 (b) 3n - 2 Ans. (a) Q10. Three integers, a, b and c, are such that a < b < c. The three integers are said to form a Pythagorean Triple is c2 = a2 + b2 or c2 - b2 = a2. For example 3, 4, 5 from a Pythagorean Triple because 52 - 42 = (5 - 4)(5 + 4) = 1 x 9 = 9 = 32 and 5, 12, 13 from a Pythagorean Triple because 132 - 122 = (13 - 12)(13 + 12) = 1 x 25 = 25 = 52 (a) In the same way, show that 7, 24 and 25 form a Pythagorean Triple. (b) Form a Pythagorean Triple (c) (i) in which the last two integers are 40 and 41, (ii) in which the first integer is 11. (i) Simplify (n + 1)2 - n2. (ii) Hence form a Pythagorean Triples in which the first integer is 101. (d) It is also possible to form Pythagorean Triple in which the last two integers differ by 2. For example 8, 15, 17 from a Pythagorean Triple because 172 - 152 = (17 - 15)(17 + 15) = 2 x 32 = 64 = 82. (i) Copy and complete the following statements: … ,35, 37 from a Pythagorean Triple because 372 – 352 = ( )( ) = ….. x ….. = ….. = ….. . 16, … , … from a Pythagorean Triple because ….. --- ….. = ( )( ) = 2 x ….. = ….. = 162. (ii) Simplify (4n2 + 1)2 - (4n2 -1)2 and hence express it as a perfect square. (iii) Form a Pythagorean Triple in which the first integer is 400 and the other two integers differ by 2. Ans. (a) 252 – 242 = (25 – 24)(25 + 24) = 1 x 49 = 49 = 72 (b) (i) The triple is 9, 40 and 41. 5 TOPIC#08 Problem Solving (c) (d) (ii) The triple is 11, 60 and 61. (i) 2n + 1 (ii) The triple is 101, 5101, 5100. (i) 12, 35, 37 form a Pythagorean Triple because 372 - 352 = (37 - 35)(37 + 35) = 2 x 72 = 144 = 122 16, 63, 65 form a Pythagorean Triple because 652 -632 = (65 - 63)(65 + 63) = 2 x 128 = 256 = 162 Q11. (ii) (4n)2 (iii) The Pythagorean Triple is 100, 39999 and 40001. Read these instructions. A Choose two different digits from 1, 2, 3, 4, 5, 6, 7, 8 and 9. B Write down the larger two-digit number which can be formed from the chosen digits. C Write down the smaller two-digit number which can be formed from the chosen digits. D Subtract the smaller number from the larger and not the result. Example: A Choose 2 and 8. B larger number is eighty-two (82). C Smaller number is twenty-eight (28). D Subtract: 82 -28 54 Result = 54 (a) The digit 3 and 7 are chosen. Follow the instructions to find the result. (b) Choose three other different pairs of digits. Follow the instructions to find the result in each case. (c) What do you notice about all these results? (d) The digit x and y, where x > , are chosen. Find expression, in terms of x and y, for the value of (i) the larger number, 6 TOPIC#08 Problem Solving (ii) Ans. the result. (a) Result = 73 - 37 = 36 (b) Choose 1 and 3. Result = 18 Choose 2 and 4. Result = 18 Choose 5 and 6. Result = 9 Q12. (c) They are all multiples of 9. (d) (i) 10x + y (ii) (10x + y) – (10y + x) = 9x - 9y (a) The first five terms of a sequence are 1, 3, 6, 10, 15. The nth term of this sequence is 1 n( n 1). 2 Find the 19th term. (b) Write down an expression, in terms of n, for the nth term of the sequence 3, 6, 10, 15, 21, …………… 190 (b) 1 (n 1)(n 2). 2 Ans. (a) Q13. The cost of parking in a car park is 10 cents for each hour. When he parked his car, John had only a large number of 10 cents coins and 20 cent coins to put into the ticket machine. The table shows how he can pay to park his car. (a) Show that there are (i) 5 ways to pay for 4 hours, 7 TOPIC#08 Problem Solving (ii) (b) 8 ways to pay for 5 hours. The table below shows the number of ways John can pay when parking for various times. Ans. (a) (i) Find the values of a and b. (ii) Write down an equation connecting x, y and z. (i) 10 then 10 then 10 then 10 20 then 10 then 10 10 then 20 then 10 10 then 10 then 20 20 then 20 (ii) 10 then 10 then 10 then 10 then 10 10 then 20 then 10 then 10 10 then 10 then 20 then 10 10 then 10 then 10 then 20 10 then 20 then 20 20 then 10 then 10 then 10 20 then 20 then 10 20 then 10 then 20 (b) (i) a = 13 (ii) x+y=z b = 21 Q14. Counters are used to make patterns as shown above. Pattern 1 contains 6 counters. 8 TOPIC#08 Problem Solving The numbers of counters needed to make each pattern form a sequence. (a) Write down the first four terms of this sequence. (b) The number of counters needed to make Pattern n is An + 2. Find the value of A. (c) Mary has 500 counters. She uses as many of these counters as she can to make one pattern. Given that this is Pattern m, find (i) the value of m, (ii) how many counters are not used. 6, 10, 14, 18 (b) A=4 (c) m = 124 Ans. (a) (i) Q15. The terms T1, T2, T3, T4, T5 of a sequence are given as follows: (ii) 2 T1 = 1 = 1 T2 = 3 = 1 + 2 T3 = 6 = 1 + 2 + 3 T4 = 10 = 1 + 2 + 3 + 4 T5 = 15 = 1 + 2 + 3 + 4 + 5 (a) (i) Write down the next two terms, T6 and T7, in the sequence 1, 3, 6, 10, 15, ……… (ii) The nth term in the sequence is given by Tn 1 n(n 1). 2 Show that this formula is true when n = 7. (iii) Use the formula to find T100. (iv) Use your answer to part (iii) to find 5 + 10 + 15 + …………+ 500. (v) Hence find the sum of all the whole numbers from 1 to 500 which are not multiplies of 5. (b) The terms S1, S2, S3, S4, S5 of a different sequence are given as follows: S1 = 1 = 1 x 1 S2 = 4 = 1 x 2 + 2 x 1 S3 = 10 = 1 x 3 + 2 x 2 + 3 x 1 S4 = 20 = 1 x 4 + 2 x 3 + 3 x 2 + 4 x 1 9 TOPIC#08 Problem Solving S5 = 35 = 1 x 5 + 2 x 4 + 3 x 3 + 4 x 2 + 5 x 1 (i) Find S6 and S7. (ii) The nth term in this sequence is given by Sn 1 n(n 1)(n 2). 6 Show that this formula is true when n = 7 (iii) (c) Find 1 x 20 + 2 x 19 + 3 x 18 + …… + 20 x 1. S2 - S1 = (1 x 2 + 2 x 1) -1 x 1 = 1 x 2 + 1 x 1 = 2 + 1 = T2 S3 - S2 = (1 x 3 + 2 x 2 + 3 x 1) - (1 x 2 + 2 x 1) = 3 + 2 + 1 = T3 Show that Ans. (a) (i) S4 - S3 = T4. (ii) Sn+1 - Sn = Tn + 1. (i) T6 = 21, (ii) if n = 7, T7 = 28 1 1 n(n 1) = (7)(7 1). 2 2 = 28 = T7 (b) (iii) T100 = 5050 (iv) (i) S6 = 56, S7 = 84 (ii) if n = 7, 25 250 1 (7)(7 1)(7 2) 6 (v) 100 000 789 6 = 84 = S7 (c) (iii) S20 = 1540 (i) S4 – S3 = 1 1 (4)(4 1)(4 2) (3)(3 1)(3 2) 6 6 = 20 – 10 = 10 12 20 = 1 2 4 (4 1) = T4 10 TOPIC#08 Problem Solving (ii) Sn + 1 – Sn = 16 (n 1)(n 2)(n 3) 16 (n)(n 1)(n 2) = 1 6 (n 1)(n 2)(n 3 n) = 1 2 (n 1)(n 2) = Tn+1 Q16. (c) The nth term of a sequence, S, is n3 +2. The first four terms are 3, 10, 29 and 66. (i) Find the fifth term of S. (ii) The first four terms of another sequence, T, are 4, 12, 32 and 70. By comparing S and T, write down (d) (a) the fifth term of T, (b) an expression, in terms of n, for the nth term of T. On Monday, two girls, Jane and Susan, collected some seashells. Jane collected x shells and Susan collected 22 more than Jane. On Tuesday, Susan gave 60 of her shells to Jane. The table shows the number of shells each girl had on the two days. Ans. (c) (d) (i) Write down an expression for y in terms of x. (ii) Given that, on Tuesday, Jane had three times as many shells as Susan, (a) write down and solve an equation in x. (b) find the total number of shells the girl collected. (i) 127 (ii) (a) (i) y = x – 38 (ii) (a) 132 (b) Tn = n3 + n + 2 x + 60 = 3y x + 60 = 3 (x – 38) 11 TOPIC#08 Problem Solving x = 87 (b) Q17. 196 A series of diagrams, using three types of triangle, is shown below. The triangles are grey, white or black. The table below shows the numbers of each type of triangle used in the diagrams. 4 n Diagram 1 2 3 Grey triangles 2 4 6 x White triangles 1 4 9 y Black triangles 0 2 6 z (a) Complete the column for Diagram 4. (b) By considering the number patterns in the table, find, in terms of n, expression for x, y and z. Ans. (a) Number of grey triangles = 8 Number of white triangles = 16 Number of black triangles = 12 (b) x = 2n y = n2 z = n2 - n .. 12