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Middle Years

8.01 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 u_n describes the nth term of the following sequences , find the indicated term:

a

u_3 in the sequence: 4, - 5 , 6, - 7 , 8, \ldots

b

u_4 in the sequence: 200, 20, 2, 0.2, 0.02, \ldots

c

u_5 in the sequence: 1, 2.5, 4, 5.5, 7, 8.5, \ldots

d

u_3 + u_5 in the sequence: 6, - 8 , 9, - 10 , 11, \ldots

e

2 u_2 - u_4 in the sequence: 9, 12, 15, 18, 21, \ldots

f

- 4 \left(u_3 + u_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 u_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

Recurrence relations
5

Determine whether the following are recurrence relations:

a

\dfrac{S_n}{n} = n + 1

b

R_n = \left( 3 R_{n - 1}\right)^{5} + R_{n - 2}

c

u_n = u_{n - 1} + 9

d

V_n = 3 \left(n - 1\right)

6

Consider the following sequence given in the table:

a

Write down the general rule for u_n in terms of n.

b

Use the general rule to find u_{20}.

n1234\ldots
u_n14916\ldots
7

Consider the following sequence given in the table:

a

Write down the general rule for u_n in terms of n.

b

Use the general rule to find u_{19}.

n1234\ldots
u_n182727\ldots
8

Write the recursive rule for each of the given description. Let u_n be the nth term.

a

To find the next term, add 5 to the previous term.

b

To find the next term, add the two previous terms.

c

To find the next term, multiply the previous term by negative 1 and then add 7.

d

To find the next term, subtract one and a half from the previous term.

e

To find the next term, subtract 6 from the previous term.

f

To find the next term, multiply the two previous terms.

g

To find the next term, multiply the previous term by negative 5 and then subtract 3.

h

To find the next term, add three and a quarter to the previous term.

9

For each of the following recursive rules, state the first 5 terms of the sequences in order:

a

u_n = 2 u_{n - 1},\ u_1 = 2

b

u_n = u_{n - 1} + 6,\ u_1 = - 8

c

b_n = - b_{n - 1} + 3,\ b_1 = 0

10

For each of the recursive relations below, find the following.

i

u_2

ii

u_3

iii

u_4

a

u_n = \left( - 1 \right)^{n + 1} u_{n - 1},\ u_1 = 2

b

u_{n + 1} = \left( 2 u_n\right)^{n - 1},\ u_1 = 3

c

u_n = n^{n + 1} + u_{n - 1},\ u_1 = 3

d

u_{n + 1} = \left( 4 u_n\right)^{n - 1} \times \left(n + 1\right),\ u_1 = 5

11

Consider the sequence which has a first term of 11 and a second term of 22, and subsequent terms are found by adding the two previous terms.

a

Write the recursive rule for the sequence, letting u_n be the n\text{th} term.

b

State the first five terms of the sequence.

Explicit rules
12

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

u_n = 8 u_{n - 1} + u_1

d

c_n = 3 n^{2}

13

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

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

14

For the following sequences:

i

State the recursive rule that describes u_n in terms of u_{n - 1}, and the initial term u_1.

ii

State the explicit rule that describes u_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
15

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

16

Consider the sequence defined by u_n = 2 u_{n - 1} + 3, u_1 = 12. Calculate the sum of the first 25 terms.

17

For each of the following pairs of sequences, compare the first 10 terms of each sequence on a graph to determine which sequence is increasing faster:

a

u_{n + 1} = u_n + 22,\ u_1 = 12

A_{n + 1} = A_n + 25,\ A_1 = 12

b

u_{n + 1} = 2.5\, u_n,\ u_1 = 5

A_{n + 1} = 3.5 A_n,\ A_1 = 5

18

For each pair of sequences, compare the first 10 terms of each sequence on a graph to determine which sequence is decreasing faster:

a

u_{n + 1} = u_n - 5,u_1 = 4 and A_{n + 1} = A_n - 10, A_1 = 12

b

u_{n + 1} = 0.2 u_n, u_1 = 20 and A_{n + 1} = 0.6 A_n, A_1 = 20

19

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

20

For the explicit rule u_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.

21

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

a

If u_n is the value of the nth term in the sequence, find u_{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.

22

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

a

Find u_{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.

23

Consider the sequence u_n = u_{n - 1} + 5, u_1 = 10

a

Find u_{40}.

b

Find the first term greater than or equal to 300.

c

If S_n is the sum of the first n terms, find S_{30}.

d

Starting from n = 1, find the minimum number of terms for the sum to be greater than 5000.

24

Consider the sequence u_{n+1} = 1.5 u_n, u_1 = 10

a

Find u_{10} to the nearest whole number.

b

Find the first term greater than or equal to 1000. Round your answer to the nearest whole number.

c

If S_n is the sum of the first n terms, find S_{18}. Round your answer to the nearest whole number.

d

Starting from n = 1, find the minimum number of terms for the sum to be greater than 50\,000.

25

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

a

Find u_{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.

26

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

a

Find u_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.

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