2025 AMC 10A Problem 11

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

Write

z = 1 + 3d

from the arithmetic sequence and

z = p^3

from the geometric one

Since

z \equiv 1 \pmod 3,

test increasing integer ratios

p

until

p^3 \equiv 1 \pmod 3

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.

Problem 11 in other years