Chapter B.1, Problem 65E

Some sequences are defined by a recursion 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 {a_n} is defined by

     $a_1=1and{a}_n=2a_{n-1}+1forn\ge 2$

then

     $\begin{array}{l}{a}_2=2a_1+1=2\cdot 1+1=3\\ a_3=2a_2+1=2\cdot 3+1=7\\ a_4=2a_3+1=2\cdot 7+1=15\end{array}$

and so on. In Problems 63–66, write the first five terms of each sequence.

65.    a₁ = 1 and a_n = 2a_{n – 1} for n ≥ 2