Maths Olympiad Prep

Track / Stage 4 / 102 of 340 #362 of 1964

Problem 362

AMC 12 late, AIME early
Algebra Difficulty 4.7 Multiple choice

Calculate:
2(2(2(2(2(2+1)+1)+1)+1)+1)+1=(). \begin{array}{l} 2(2(2(2(2(2+1)+1)+1)+1)+1)+1 \\ =(\quad) . \end{array}

Pick one

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Official solution

1. C.
 Let S=2(2(2(2(2(2+1)+1)+1)+1)+1)+1.  Then S+1=26+25+24+23+22+2+1+1=27⇒S=27−1=127.  \begin{array}{l} \text { Let } S=2(2(2(2(2(2+1)+1)+1)+1)+1)+1 \text {. } \\ \text { Then } S+1=2^{6}+2^{5}+2^{4}+2^{3}+2^{2}+2+1+1=2^{7} \\ \Rightarrow S=2^{7}-1=127 \text {. } \end{array}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.