2017 AMC 10A Problems/Problem 1

Problem

What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$ ?

$\textbf{(A)}\ 70\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 127\qquad\textbf{(D)}\ 159\qquad\textbf{(E)}\ 729$

Solution

Notice this is the term $a_6$ in an arithmetic sequence, defined recursively as $a_1 = 3, a_n = 2a_(n-1) + 1.$ Thus: $\begin{split} a_2 = 3*2 + 1 = 7.\\ a_3 = 7 *2 + 1 = 15.\\ a_4 = 15*2 + 1 = 31.\\ a_5 = 31*2 + 1 = 63.\\ a_6 = 63*2 + 1 = \boxed{\textbf{(C)}\ 127.} \end{split}$

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2017 AMC 10A Problems/Problem 1

Problem

Solution