2022 AMC 12A Problem 16

Below is the professionally curated solution for Problem 16 of the 2022 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12A solutions, or check the answer key.

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Concepts:triangular number perfect square recursion

Difficulty rating: 1800

16.

A triangular number is a positive integer that can be expressed in the form $t_n=1+2+3+\cdots+n,$ for some positive integer $n.$ The three smallest triangular numbers that are also perfect squares are $t_1=1=1^2,$ $t_8=36=6^2,$ and $t_{49}=1225=35^2.$ What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?

$6$

$9$

$12$

$18$

$27$

Solution:

Square triangular numbers satisfy the recurrence $N_k=34N_{k-1}-N_{k-2}+2.$ Starting from $N_2=36$ and $N_3=1225,$ the next term is $N_4=34\cdot1225-36+2=41616.$

Indeed $41616=204^2=\dfrac{288\cdot289}{2},$ so it is both a perfect square and $t_{288}.$

The sum of its digits is $4+1+6+1+6=18.$

Thus, the correct answer is D.

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