AMC 10 · 2005 · #13

Grade 11 algebra
Archetype Telescoping Collapse └ Sub-archetype Product Ratio Cancellation
exponentslogarithm-propertiespattern-recognition easier-related-problempattern-recognition ↑ Prerequisites: exponentslogarithm-properties
📏 Medium solution 💡 2 insights
Problem
A chain of exponential equations runs from base four up to one hundred twenty-eight, each using the previous result as its new base. Find the product of all the exponents.

Pick an answer.

(A)
2
(B)
$\frac{5}{2}$
(C)
3
(D)
$\frac{7}{2}$
(E)
4
How to solve
Strategy Solve an Easier Related Problem

A product of 124 unknown irrational numbers cannot be computed one factor at a time, and none of them has a nice closed form on its own. But a two-link version of the same chain is easy, and it reveals that stacking the equations multiplies the exponents. Extending that pattern collapses the whole chain to a single equation relating its two endpoints, 4 and 128.

1STEP 1

Shrink to two links

Substituting merges two links and multiplies their exponents.

6 = 5^x₂ = (4^x₁)^x₂ = 4^x₁ x₂
2STEP 2

Extend down the chain

The same move works at every link, so the whole chain collapses.

4^x₁ x₂ … x_n = n+4 ⟹ 4^x₁ x₂ … x_n+1 = (n+4)^x_n+1 = n+5
3STEP 3

Count the links and land on 128

Counting the links lands the right side on 128.

n = 124: 4^ x₁ x₂ … x₁₂₄ = 128, P = x₁ x₂ … x₁₂₄
4STEP 4

Put both sides in base 2

Both sides are powers of two, so the exponents can be compared.

4^P = (2²)^P = 2²P = 2⁷ = 128
5STEP 5

Read off the exponent

Reading them off gives 7/2, choice (D).

2P = 7 ⟹ P = 7/2
Answer
7/2
Bound the answer without solving. The product satisfies 4^P = 128, and 4³ = 64 ＜ 128 ＜ 256 = 4⁴, so 3 ＜ P ＜ 4. That single inequality eliminates 2, 5/2, 3, and 4 outright and leaves only 7/2. A second check on the shape: each x_k = log_k(k+1) is slightly larger than 1, so the product must exceed 1 but grow only slowly across 124 factors, which fits a value near 3.5 rather than anything large.
💡Key takeaway

When each power feeds into the next as a base, the exponents multiply, so a long chain collapses to a single comparison between its two ends.

  • Shrink to two links
  • Extend down the chain
  • Count the links and land on 128
  • Put both sides in base 2
  • Read off the exponent

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