 iGCSE (2021 Edition)

9.01 Sequences

Worksheet
Sequences
1

For each of the following sequences, find the indicated term:

a

Third term in the sequence: 2, - 4 , 6, - 8 , 16, \ldots

b

Fourth term in the sequence: 3, 3.5, 4, 4.5, 5, 5.5, \ldots

c

Fifth term in the sequence: 5, 4, 3, 2, 1, 0, - 1 , \ldots

2

If T_n describes the nth term of the following sequences , find the indicated term:

a

T_3 in the sequence: 4, - 5 , 6, - 7 , 8, \ldots

b

T_4 in the sequence: 200, 20, 2, 0.2, 0.02, \ldots

c

T_5 in the sequence: 1, 2.5, 4, 5.5, 7, 8.5, \ldots

d

T_3 + T_5 in the sequence: 6, - 8 , 9, - 10 , 11, \ldots

e

2 T_2 - T_4 in the sequence: 9, 12, 15, 18, 21, \ldots

f

- 4 \left(T_3 + T_4\right) in the sequence: 1, 4, 5, 9, 14, 23, \ldots

3

In the sequence 5, - 5 , 7, - 7 , 9, - 9 , \ldots , find n if T_n = 7.

4

For each of the following sequences:

i

Describe the recurring pattern in words.

ii

Find the next number in the sequence.

a

- 1 , 1, 3, 5, 7, \ldots

b

64, 32, 16, 8, 4, 2, \ldots

c

2, - 4 , 6, - 8 , 10, - 12 , \ldots

Explicit rules
5

Determine whether the following are explicit relations:

a

b_n = b_{n - 1} b_{n - 2}

b

d_n = n^{2} + 3 n + 8

c

a_n = 8 a_{n - 1} + a_1

d

c_n = 3 n^{2}

6

For each of the following explicit rules, state the first 5 terms of the sequences in order starting with n=1:

a

b_n = 5 n - 2

b

s_n = n^{2} + 6

c

t_n = 2 n^{2} + n - 3

7

For the following sequences:

i

State the recursive rule that describes T_n in terms of T_{n - 1}, and the initial term T_1.

ii

State the explicit rule that describes T_n in terms of n.

a

A sequence starts with a first term of 1300 and each subsequent term increases by 2.5\% of the previous term.

b

A sequence starts with a first term of 44 and each term is 77 more than the previous term.

Technology
8

Find the indicated term in the following sequences:

a

21st term in the sequence: - 3 , - 6 , - 9 , - 12 , - 15 , \ldots

b

53rd term in the sequence: 2, 3.5, 5, 6.5, 8, 9.5, \ldots

c

39th term in the sequence: 10, 8, 6, 4, 2, 0, - 2 , \ldots

9

For the explicit rule T_n = 0.7 n - 5 which starts at n = 1, find the sum of the first 50 terms.

10

For the explicit rule T_n = \left( 0.2 n\right)^{2} + 5 n which starts at n = 1, find the sum of the terms from the 20th to the 30th term inclusive.

11

Consider the sequence 3, 7, 11, 15, 19, 23, \ldots

a

If T_n is the value of the nth term in the sequence, find T_{10}.

b

Find the first term greater than 100.

c

If S_n is the sum of the first n terms, find S_{10}.

d

Starting from n = 1, find the minimum number of terms for the sum to first exceed 500.

12

Consider the sequence 1, 2, 4, 8, 16, \ldots

a

Find T_{13}.

b

Find the first term greater than 10\,000.

c

If S_n is the sum of the first n terms, find S_{20}.

d

Starting from n = 1, find the greatest number of terms such that the sum is still less than 10\,000\,000.

13

Consider the sequence T_n = 5 - 3 n, starting at n = 1:

a

Find T_{30}.

b

Find the first term less than - 150.

c

If S_n is the sum of the first n terms, find S_{15}.

d

Starting from n = 1, find the minimum number of terms required for the sum to be less than - 600.

14

Consider the sequence T_n = 200 \times 0.5^{n - 1}

a

Find T_8.

b

Find the first term less than 0.5.

c

If S_n is the sum of the first n terms, find S_{10}.

d

Starting from n = 1, find the minimum number of terms for the sum to first exceed 399.9.