Exercise 59 Page 320

We are asked to write a recursive rule for the sequence.

3, 7, 15, 31, 63, \dots

Let's look for a pattern in the difference of the consecutive terms.

We see that each difference is a power of

2 .

Consider the difference between the fourth and the third term,

2

and

4 .

a_{4} = a_{3} + 2^{4} \Leftrightarrow a_{4} = a_{4 - 1} + 2^{4}

We can write the recursive equation.

a_{n} = a_{n - 1} + 2^{n}

The first term and the recursive equation gives us the recursive rule.

Recursive Rule : a_{1} = 3, a_{n} = a_{n - 1} + 2^{n}