Exercise 42 Page 279

Let's start by recalling the recursive formula for an arithmetic sequence.

A (n) = A (n - 1) + d; A (1) = A_{1}

In the above formula,

n

is the number of the term and

d

is the common difference. Looking at the given explicit formula, we can identify the common difference

d

and the first term

A_{1} .

General Rule : Given Rule : A (n) = A_{1} + (n - 1) d A (n) = 5 + (n - 1) (3)

From the explicit formula, we can see that

3

is the common difference and that the first term is

5 .

Now we have enough information to form a recursive formula for this sequence.

A (n) = A (n - 1) + 3; A (1) = 5

Note that, when writing the recursive formula for a specific arithmetic sequence, we need to include the first term as a separate piece of information. If the first term is not stated, the sequence could start on any value and increase by the given common difference. This means that the sequence would not necessarily match our sequence.