Question

The sequence $1, x, y, z$ is arithmetic. The sequence $1, p, q, z$ is geometric. Both sequences are strictly increasing and contain only integers, and $z$ is as small as possible. What is the value of $x + y + z + p + q?$

$66$

$91$

$103$

$132$

$149$

Answer: E

Concepts:arithmetic sequence geometric sequence modular arithmetic

Difficulty rating: 1500

Solution:

From the arithmetic sequence, $z = 1 + 3d,$ so $z \equiv 1 \pmod 3.$ From the geometric one, $z = p^3$ for some integer ratio $p \ge 2.$ We want the smallest such $z,$ so test $p = 2, 3, 4.$ Only $p = 4$ works, since $p^3 = 64 \equiv 1 \pmod 3.$ That forces $d = 21,$ and the sequences are $1, 22, 43, 64$ and $1, 4, 16, 64.$ So $x + y + z + p + q$ $= 22 + 43 + 64 + 4 + 16$ $= 149.$ Thus, E is the correct answer.

2025 AMC 10A Problem 11

Answer: E

Difficulty rating: 1500

Solution:

Problem 11 in Other Years