Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.3 AMC 10/12 Find the answer

Given an arithmetic-geometric sequence $\{a\_n\}$ , the sum of its first $n$ terms is $10$ , and the sum of its first $2n$ terms is $30$ . Find the sum of its first $3n$ terms.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

By the properties of the sum formula for an arithmetic-geometric sequence, we know that $S_{10}$ , $S_{20}-S_{10}$ , and $S_{30}-S_{20}$ form an arithmetic-geometric sequence.

Thus, we have:
$(30-10)^2 = 10 \times (S_{30} - 30)$

Solving for $S_{30}$ , we get:
$S_{30} = 70$

Therefore, the sum of the first $3n$ terms is $\boxed{70}$ .

The properties of the sum formula for an arithmetic-geometric sequence tell us that $S_{10}$ , $S_{20}-S_{10}$ , and $S_{30}-S_{20}$ form an arithmetic-geometric sequence, which leads us to the result.

This problem tests your understanding of the sum formula for an arithmetic-geometric sequence and its properties, as well as your reasoning and computational skills. It is a moderately difficult question.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.