Exercise 59 Page 320

We are asked to write a recursive rule for the sequence. 3, 7, 15, 31, 63, ... 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 terms, 2 and 4. a_4=a_3+2^4 ⇔ 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. \begin{gathered} \textbf{Recursive Rule: } a_1=3, \ a_n=a_{n-1}+2^n \end{gathered}