2017 AMC 10A Problems/Problem 1
What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$?
Solution 1
Notice this is the term $a_6$ in a recursive sequence, defined recursively as $a_1 = 3, a_n = 2a_{n-1} + 1.$ Thus: [\begin{split} a_2 = 32 + 1 = 7.\ a_3 = 7 2 + 1 = 15.\ a_4 = 152 + 1 = 31.\ a_5 = 312 + 1 = 63.\ a_6 = 63*2 + 1 = \boxed{\textbf{©}\ 127} \end{split}]$\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{©}\ 127\qquad\textbf{(D)}\ 159\qquad\textbf{(E)}\