2002 AMC 10B Problem 23

Problem 23 of 25HarderAlgebra Number Theory

Let

\{a_k\}

be a sequence of integers such that

a_1 = 1

and

a_{m+n} = a_m + a_n + mn

for all positive integers

m

and

n.

What is

a_{12}?

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Solution

Setting

n = 1,

we get

a_{m+1} = a_m + a_1 + m

= a_m + (m + 1),

a_{m+1} - a_m = m + 1.

Summing from

m = 1

11,

\begin{aligned} a_{12} - a_1 &= 2 + 3 + \cdots + 12 \\ &= \dfrac{12\cdot 13}{2} - 1 \\ &= 77. \end{aligned}

Therefore

a_{12} = 1 + 77 = 78.

Thus, the correct answer is D.

Tagged: functional equation · telescoping · triangular number