A sequence of numbers begins with t_1 = 4. For all integers n \geq 2, each term is defined… — GRE General Test

A sequence of numbers begins with $t_1 = 4$ . For all integers $n \geq 2$ , each term is defined by $t_n = 3t_{n-1} - 2$ . What is the value of $t_5$ ?

Answer: 244

Answer

The 5th term of the sequence,

t_5

, is equal to 244.

Applying the recursive relation

t_n = 3t_{n-1} - 2

step-by-step starting from

t_1 = 4

gives

t_2 = 10

t_3 = 28

t_4 = 82

, and finally

t_5 = 244

Step-by-Step Solution

Find the second term,

t_2

, using

t_1 = 4

t_2 = 3(4) - 2 = 12 - 2 = 10

Substitute

n = 2

into the recursive rule

t_n = 3t_{n-1} - 2

Find the third term,

t_3

, using

t_2 = 10

t_3 = 3(10) - 2 = 30 - 2 = 28

Substitute

n = 3

into the recursive rule.

Find the fourth term,

t_4

, using

t_3 = 28

t_4 = 3(28) - 2 = 84 - 2 = 82

Substitute

n = 4

into the recursive rule.

Find the fifth term,

t_5

, using

t_4 = 82

t_5 = 3(82) - 2 = 246 - 2 = 244

Substitute

n = 5

into the recursive rule.

Key Concept

Evaluating terms in a recursively defined sequence

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