A sequence of numbers t_1, t_2, t_3, \dots is defined by t_1 = 3 and t_{n+1} = 3t_n - 2^n … — ACT

A sequence of numbers $t_1, t_2, t_3, \dots$ is defined by $t_1 = 3$ and $t_{n+1} = 3t_n - 2^n$ for all integers $n \geq 1$ . What is the value of the fourth term, $t_4$ ?

A
5
B
9
43Answer
D
62
E
119

Answer

The value of the fourth term is 43.

To find

t_4

, we use the recursive formula

t_{n+1} = 3t_n - 2^n

. Substituting

n=1

gives

t_2 = 3(3) - 2 = 7

. Substituting

n=2

gives

t_3 = 3(7) - 4 = 17

. Substituting

n=3

gives

t_4 = 3(17) - 8 = 43

Step-by-Step Solution

Calculate the second term,

t_2

, by substituting

n = 1

and

t_1 = 3

into the formula

t_{n+1} = 3t_n - 2^n

t_2 = 3t_1 - 2^1 = 3(3) - 2 = 7

To progress to the fourth term, we must first find each preceding term in the sequence.

Calculate the third term,

t_3

, by substituting

n = 2

and

t_2 = 7

into the formula.

t_3 = 3t_2 - 2^2 = 3(7) - 4 = 17

Using the value of the second term allows us to find the third term.

Calculate the fourth term,

t_4

, by substituting

n = 3

and

t_3 = 17

into the formula.

t_4 = 3t_3 - 2^3 = 3(17) - 8 = 43

This completes the recursive process to find the target term.

Key Concept

Evaluating terms of a sequence defined by a recursive formula.

Estimated Time:1m 30s

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