Maths Olympiad Prep

Track / Stage 3 / 157 of 260 #157 of 1964

Problem 157

AMC 10/12, early questions
Algebra Difficulty 3.4 Find the answer

Given (1+x)(2−x)6=a0+a1(x−1)+a2(x−1)2+…+a7(x−1)7(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{7}(x-1)^{7}, find the value of a3a_{3}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

Since (1+x)(2−x)6=[(x−1)+2](1−x+1)6=[(x−1)+2][(x−1)−1]6(1+x)(2-x)^{6}=[(x-1)+2](1-x+1)^{6}=[(x-1)+2][(x-1)-1]^{6},
we can expand it as [(x−1)+2][(60)(x−1)6−(61)(x−1)5+(62)(x−1)4−(63)(x−1)3+(64)(x−1)2−(65)(x−1)+(66)][(x-1)+2][\binom{6}{0}(x-1)^{6}-\binom{6}{1}(x-1)^{5}+\binom{6}{2}(x-1)^{4}-\binom{6}{3}(x-1)^{3}+\binom{6}{4}(x-1)^{2}-\binom{6}{5}(x-1)+\binom{6}{6}],
and given (1+x)(2−x)6=a0+a1(x−1)+a2(x−1)2+…+a7(x−1)7(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{7}(x-1)^{7},
thus, a3=−2(63)+(64)=−25a_{3}=-2\binom{6}{3}+\binom{6}{4}=-25,
so the answer is −25\boxed{-25}.
By transforming the left side of the equation into [(x−1)+2][(x−1)−1]6[(x-1)+2][(x-1)-1]^{6} and then expanding it according to the binomial theorem, we can find the coefficient of (x−1)3(x-1)^{3}, which is the value of a3a_{3}.
This problem mainly examines the application of the binomial theorem and the general formula for the expansion of a binomial series, and is considered a medium-level question.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.