Download Number Sequences1

Number Sequences
In the sequence 2, 4, 6, 8, 10... there is an obvious pattern. Such sequences can
be expressed in terms of the nth term of the sequence. In this case, the nth term
= 2n. To find the 1st term, put n = 1 into the formula, to find the 4th term, replace
the n's by 4's: 4th term = 2 × 4
= 8.
Example:
What is the nth term of the sequence 2, 5, 10, 17, 26... ?
n
=
1
2
3
4
5
n² =
1
4
9
16
25
n² + 1 = 2
5
10
17
26
This is the required sequence, so the nth term is n² + 1. There is no easy way of
working out the nth term of a sequence, other than to try different possibilities.
Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably
be a three in the formula, etc.
In many cases, square numbers will come up, so try squaring n, as above. Also,
the triangular numbers formula often comes up. This is n(n + 1)/2 .
Example:
Find the nth term of the sequence: 2, 6, 12, 20, 30...
n
= 1 2 3 4 5
n(n + 1)/2 = 1 3 6 10 15
Clearly the required sequence is double the one we have found the nth term for,
therefore the nth term of the required sequence is 2n(n+1)/2 = n(n + 1).
The Fibonacci sequence is an important sequence which is as follows: 1, 1, 2, 3,
5, 8, 13, 21, ... . The next term of this well-known sequence is found by adding
together the two previous terms.
Practise Exam Paper
1. The distance between two points on a map is 5 cm correct to the nearest
centimetre.
(a) Write down the (i) least upper bound of the measurement (ii) greatest lower
bound of the measurement.
(b) The scale of the map is 1 to 20 000. Work out the actual distance in real life (in
kilometres) between the upper and lower bounds.
4 marks
2. Triangle ABC is isosceles. AB = AC = 12cm . Angle ABC is 55 degrees. Calculate
the area of the triangle correct to 3 significant figures.
3. Solve the equations:
(i) 4y² - 81 = 0
(ii) 1 + 1 = -1
(x + 2)
3
6 marks
4. There are 8 eggs. Two of the eggs have passed their sell by date and are 'bad'. 3
eggs are selected at random.
(a) Complete the probability tree diagram.
(b) Work out the probability that 3 'good' eggs will be selected.
(c) Work out the probability that at least one 'bad' egg will be selected.
9 marks
5. (a) Solve the inequality 7x + 3 > 17 + 5x
(b) Simplify the following.
(i) 2x³ × 6x²
(ii) (3y³)²
(c) Multiply out and simplify (2x - 1)(x - 3)
6 marks
6.
(a) Write down the nth term of the sequence 2, 5, 8, 11, ... .
(b) Write down the nth term of the sequence 2, 5, 10, 17, ... .
4 marks
7. Twenty five people took a test. The points scored are grouped in the frequency
table below.
(a) Work out an estimate for the mean number of points scored.
Points
Scored
Number of
people
1 to 5
1
6 to 10
2
11 to 15
5
16 to 20
7
21 to 25
8
26 to 30
2
(b) Complete the table below to show the cumulative frequency for this data.
(c) Draw a cumulative frequency graph for this data.
(d) Use your graph to find an estimate for the median of this data.
Points
Scored
1 to 5
1 to 10
1 to 15
1 to 20
1 to 25
1 to 30
Cumulative
frequency
8.
AB : AC = 1 : 3
(i) Work out the length of CD.
(ii) Work out the length of BC.
4 marks
9. Matthew and Nicola divide £94.50 in the ratio 11 : 4. How much does each of
them receive?
3 marks
10. (a) 12 can be written in the form ab where a and b are prime numbers.
Calculate the values of a and b.
(b) B = 12 + 3 . Without using your calculator show that B² = 27.
4 marks
11. The temperature from a factory furnace varies inversely as the square of the
distance from the furnace.
The temperature 2 metres from the furnace is 50 degrees Celsius.
Calculate the temperature 3.5 metres from the furnace. Give your answer to 2
decimal places.
5 marks
12. A planet is 81 900 000 000 000 km from the Earth.
(a) Write 81 900 000 000 000 in standard form.
Light travels 3 × 10^5 km in 1 second.
(b) Calculate the number of seconds that light takes to travel from the planet to the
Earth. Give your answer in standard form correct to 2 significant figures.
(c) Convert your answer to part (b) to days. Give your answer as an ordinary
number.
7 marks
13. Triangle ABC and vectors a and b are shown on the grid.
(a) Draw the position of the triangle ABC after translation by the vector b - 2a.
(b) (i) Write the vector AB in terms of a and b.
(ii) Write the vector BC in terms of a and b.
(c) D is an unmarked point on the grid. BD = 2/3 BC. AD = xa + yb . Use your
answers to (b) to calculate the values of x and y. You must show all your working.
7 marks
14. A company makes compact discs (CDs).
The total cost, P pounds, of making n compact discs is given by the formula P = a +
bn , where a and b are constants.
The cost of making 1000 compact discs is £58 000.
The cost of making 2000 compact discs is £64 000.
Calculate the values of a and b.
4 marks