AMC 10 · 2017 · #1

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
order-of-operationsexponentspattern-recognition identify-subproblemspattern-recognition ↑ Prerequisites: order-of-operations
📏 Medium solution 💡 2 insights
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Problem
Evaluate the nested expression (2(2(2(2(2(2+1)+1)+1)+1)+1)+1).

Pick an answer.

(A)
70
(B)
97
(C)
127
(D)
159
(E)
729

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The expression is one big calculation built from many identical small ones: every layer is just "double the inside, then add 1." Tool #7 (Identify Subproblems) lets us peel the parentheses one at a time, innermost first, turning a scary tower into six easy steps. Tool #5 (Look for a Pattern) reveals that each running total is one less than a power of 2, and Tool #3 (Eliminate Possibilities) uses that pattern to lock onto the one choice that fits.

1STEP 1

Read the nesting inside-out

The parentheses nest inside one another, so start at the center (2+1) and work outward, each shell doubling the inside then adding 1.

( 2(…)+1 ) means: double the inside, then add 1
2STEP 2

Double-and-add, first three layers

Start at the center: 2+1=3. Then double-and-add outward: 2 × 3 + 1 = 7, then 2 × 7 + 1 = 15.

2+1=3, 2(3)+1=7, 2(7)+1=15
3STEP 3

Finish the remaining layers

Same rule for the last three shells: 2 × 15 + 1 = 31, 2 × 31 + 1 = 63, 2 × 63 + 1 = 127 — the whole expression.

2(15)+1=31, 2(31)+1=63, 2(63)+1=127
4STEP 4

Pattern check picks the answer

The totals 3, 7, 15, 31, 63, 127 are each one less than a power of 2, so the value is 2⁷ - 1 = 127 — only (C) fits.

3,7,15,31,63,127 = 2²-1, 2³-1,…,2⁷-1 → 127 = (C)
Answer
127
Two independent routes agree on 127: direct inside-out computation (3 → 7 → 15 → 31 → 63 → 127) and the pattern 2⁷ - 1. The result sits between the choices 97 and 159, which is sensible for an expression whose biggest step is 2 × 63 + 1, and it matches (C).
💡Key takeaway

When parentheses are stacked inside each other, open them from the inside out — one easy "double and add one" step at a time.

  • Read the nesting inside-out
  • Double-and-add, first three layers
  • Finish the remaining layers
  • Pattern check picks the answer

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