Chapter B.1, Problem 65E

Some sequences are defined by a recursive formula- that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if $\left\{a_n\right\}$ is defined by

$a_1=1\text{ and }{a}_n=2a{n-1}_{}+1\text{ for }n\ge 2$

Then

$a_2=2a_1+1=2\cdot 1+1=3$

$a_3=2a_2+1=2\cdot 3+1=7$

$a_4=2a_3+1=2\cdot 7+1=15$

and so on. In Problem $63\text{-}66$, write the first five terms of each sequence.

$a_1=1\text{ and }{a}_n=2a{n-1}_{}\text{ for }n\ge 2$