A sequence a_1, a_2, a_3, \dots, a_n is defined by a_1 = 1 and a_n = 2a_{n-1} + 3 for all … — GRE General Test

A sequence $a_1, a_2, a_3, \dots, a_n$ is defined by $a_1 = 1$ and $a_n = 2a_{n-1} + 3$ for all integers $n \geq 2$ . What is the value of $a_4$ ?

The value of

a_4

29

Using the recursive rule

a_n = 2a_{n-1} + 3

starting with

a_1 = 1

, we compute

a_2 = 2(1) + 3 = 5

a_3 = 2(5) + 3 = 13

, and finally

a_4 = 2(13) + 3 = 29

Calculate the second term

a_2

using the recursive formula with

a_1 = 1

a_2 = 2(1) + 3 = 5

The recursive rule requires the previous term to compute the next term.

Calculate the third term

a_3

using

a_2 = 5

a_3 = 2(5) + 3 = 13

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 3

Calculate the fourth term

a_4

using

a_3 = 13

a_4 = 2(13) + 3 = 29

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 4

Recursive Sequences

Question