AMC 10 · 2022 · #9

Grade 8 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint

exponentssequences-arithmeticextreme-principleoptimizationguess-and-check extreme-principleguess-and-checkeasier-related-problemsystematic-enumeration ↑ Prerequisites: exponentssequences-arithmetic

📏 Medium solution 💡 2 insights

Problem

A strictly increasing arithmetic sequence of positive integers is given. For its eighth term, two raised to that term equals two to the twenty-seventh times the term itself. Find the smallest value the third term can take.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Extreme Principle

The problem splits cleanly into two halves. First, pin down a₇: Tool #9 (Easier Related Problem) turns the ugly 2^a₇ = 2²⁷ · a₇ into the single clean relation 2^a₇ - 27 = a₇ by dividing out 2²⁷; Tool #4 (Introduce a Variable) names the exponent k = a₇ - 27 so the equation becomes 2^k = k + 27; Tool #6 (Guess and Check) settles that equation with a six-row table, since doubling outruns adding 1 almost immediately. Second, minimise a₂: Tool #14 (Extreme Principle) is the whole point — with a₇ nailed to a single number, a₂ = a₇ - 5d shrinks as d grows, so push d to the largest value the constraint "a₀ is still positive" allows. Tool #3 (Eliminate Possibilities) confirms at the end that the other four choices correspond to a d that is either too small or an off-by-one in the index.

1STEP 1

Divide out the common base

Cancel the common base.

2^a₇ = 2²⁷ · a₇ ⟺ 2^a₇ - 27 = a₇

2STEP 2

Force a power of two

The term must be a power of two.

k = a₇ - 27 ≥ 1, a₇ = 2^k, a₇ = k + 27 ⟹ 2^k = k + 27

3STEP 3

Solve for the exponent

Check the small exponents one at a time.

k & 1 & 2 & 3 & 4 & 5 & 6 ; 2^k & 2 & 4 & 8 & 16 & 32 & 64 ; k+27 & 28 & 29 & 30 & 31 & 32 & 33 ⟹ k = 5, a₇ = 32

4STEP 4

Write the third term via the difference

Express it via the common difference.

a₂ = a₇ - 5d = 32 - 5d, a₀ = a₇ - 7d = 32 - 7d, d ∈ {1, 2, 3, …}

5STEP 5

Push the difference to its limit

A positive first term caps the difference.

32 - 7d ≥ 1 ⟹ d ≤ 31/7 ⟹ d_max = 4

6STEP 6

Read the minimum

The minimum is 12.

d = 4: (a₀, …, a₇) = (4, 8, 12, 16, 20, 24, 28, 32), a₂ = 32 - 5 · 4 = 12 → (B)

Answer

Two independent checks. First, a₇ = 32 satisfies the original equation exactly: 2³² = 2²⁷ · 32 = 2²⁷ · 2⁵, true. Second, sweep every legal common difference and list the resulting a₂ = 32 - 5d: d = 1 gives 27, d = 2 gives 22, d = 3 gives 17, d = 4 gives 12, and d = 5 is illegal because it would make a₀ = 32 - 35 = -3, not a positive integer. The smallest surviving value is 12, and it is the only one of {27, 22, 17, 12} that is also the smallest, so nothing was missed. It is also a good sign that 12 sits in the middle of the choice list rather than at an end — the choices 17 and 22 are exactly the answers a solver gets by giving up on d one or two steps early.

💡Key takeaway

Cancel the shared 2²⁷ and the equation forces a₇ = 32; after that, making a₂ as small as possible just means taking the biggest steps the sequence can afford before its first term drops below 1.

Divide out the common base
Force a₇ to be a power of 2
Solve 2^k = k + 27
Write a₂ in terms of d
Push d to its extreme
Read off the minimum