2017 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 1 Easy

What is the value of (2(2(2(2(2(2 + 1) + 1) + 1) + 1) + 1) + 1)? (2017 AMC 10A Problem, Question#1)

  • A.

    70

  • B.

    97

  • C.

    127

  • D.

    159

  • E.

    729

Answer:C

Notice this is the term a_6 in a recursive sequence, defined recursively

a_2=3*2+1=7.

a_3=7*2+1=15.

a_4=15 *2+1=31.

a_5=31*2+1=63.

asa_1=3, a_n=2a_{n-1}+1.Thus: a_{6}=63*2+1= (\rm C)127.

Starting to compute the inner expressions, we see the results are 1, 3, 7, 15,\ldots. This is always 1 less than a power of 2. The only admissible answer choice by this ruleis thus(\rm C) 127.

Working our way from the innermost parenthesis outwards and directly computing, we have (\rm C)127.

If you distribute this you get a sum of the powers of 2. The largest power of 2 in the series is 64, so the sumis (\rm C) 127.

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