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2004 AMC 10A Problem 18

2004 AMC 10A Problem 18AMC 10 Step by Step

Name: Date: September 13, 2026

Sequences & Series · difficulty 3 of 5 · about 4 min · position P16-20

A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

A)
$1$
B)
$4$
C)
$36$
D)
$49$
E)
$81$

Answer

Key insight

Terms are 9, 11 + d, 29 + 2d; the geometric condition gives d = 10 or d = -14, and d = -14 yields third term 1.

Solution

Let the common difference be $d$ , so the arithmetic progression is $9,\ 9 + d,\ 9 + 2d$ . After the additions the numbers are
$9,\qquad 11 + d,\qquad 29 + 2d.$

Three numbers $a, b, c$ form a geometric progression exactly when $b^2 = ac$ . Hence
$(11 + d)^2 = 9(29 + 2d).$
Expanding: $d^2 + 22d + 121 = 261 + 18d$ , so $d^2 + 4d - 140 = 0$ , which factors as $(d + 14)(d - 10) = 0$ .

$d = 10$ : the geometric progression is $9, 21, 49$ (ratio $\tfrac73$ ), third term $49$ .
$d = -14$ : the geometric progression is $9, -3, 1$ (ratio $-\tfrac13$ ), third term $1$ .

Both are valid, and the smaller third term is $1$ .

The answer is $\boxed{\textbf{(A)}\ 1}$ .

Why this works

Parametrize the arithmetic progression by its difference and translate "geometric" into the single equation $b^2 = ac$ ; that always produces a quadratic in $d$ . Nothing in the problem forbids a negative difference or a negative ratio, so both roots must be examined. The word "smallest possible" is a hint that more than one progression works.

Alternative approach

Work from the answer choices: the third term $29 + 2d$ equals $1$ when $d = -14$ , giving the sequence $9, -3, 1$ , which is geometric since $(-3)^2 = 9 \cdot 1$ . Choice (A) is the smallest option and it works, so no further checking is needed.

The trap

Discarding d = -14 because it makes the sequence decreasing or the ratio negative, and answering 49.

Common mistakes

Discarding d = -14 because it makes the sequence decreasing or the ratio negative, and answering 49.
Adding $2$ and $20$ to the wrong terms, or forgetting that the third arithmetic term is $9 + 2d$ , not $9 + d$ .

Techniques

Apply an identity: SFFT, sum of squares, difference of cubes, Vieta · Set up the equation/formula and compute; no special trick needed