AMC 10 · 2016 · #16

Grade 8 arithmetic
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
sequences-geometricoptimizationquadratic-equations convert-to-algebra ↑ Prerequisites: sequences-geometric
📏 Medium solution 💡 3 insights
Problem
An infinite geometric series adds up to a positive number S. Its second term equals 1. Find the smallest value S can possibly be.

Pick an answer.

(A)
$\frac{1+\sqrt{5}}{2}$
(B)
2
(C)
$\sqrt{5}$
(D)
3
(E)
4

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for a smallest possible value, which is a minimization problem, so the Extreme Principle leads. To use it, name the first term and ratio as variables, turn the two facts into algebra, and write S as a single expression in r. Then S is smallest exactly when one denominator quantity is largest, and that maximum is easy to find.

1STEP 1

Write the sum with a and r

Let a be the first term and r the ratio; an infinite geometric series sums to a divided by (1 minus r), valid only when |r| ＜ 1.

S = a/(1-r), |r| ＜ 1
2STEP 2

Use the second term to cut a variable

The second term is a times r, so a times r = 1, giving a = 1/r; substituting makes S = 1/(r − r²), one variable.

a r = 1 → a = 1/r, S = (1/r)/(1-r) = 1/(r - r²)
3STEP 3

Smallest S means largest denominator

Since S ＞ 0 its denominator r − r² must be positive, so 0 ＜ r ＜ 1; a bigger denominator under a fixed numerator gives a smaller S.

S = 1/(r - r²) is smallest when r - r² is largest, 0 ＜ r ＜ 1
4STEP 4

Maximize the denominator

Complete the square: r − r² = 1/4 − (r − 1/2)² peaks at 1/4 when r = 1/2, so S ≥ 4 is the minimum, answer (E).

r - r² = 1/4 - (r - 1/2)² ≤ 1/4 → S ≥ 1/(1/4) = 4
Answer
4
Plug r = 1/2 back in: first term a = 1/r = 2, second term = 2 * 1/2 = 1 as required, and the sum is 2 / (1 - 1/2) = 4. Any other allowed r between 0 and 1 makes r - r² smaller than 1/4, which makes S larger than 4, so 4 is genuinely the minimum and matches answer (E).
💡Key takeaway

To make a fraction as small as possible, make its bottom as big as possible.

  • Write the sum with a and r
  • Use the second term to cut a variable
  • Smallest S means largest denominator
  • Maximize the denominator

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