2021 AMC 12A Spring Problem 9

Below is the professionally curated solution for Problem 9 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:difference of squares telescoping

Difficulty rating: 1560

Which of the following is equivalent to $(2 + 3)(2^2 + 3^2)(2^4 + 3^4)(2^8 + 3^8)(2^{16} + 3^{16})(2^{32} + 3^{32})(2^{64} + 3^{64})?$

$3^{127} + 2^{127}$

$3^{127} + 2^{127} + 2 \cdot 3^{63} + 3 \cdot 2^{63}$

$3^{128} - 2^{128}$

$3^{128} + 2^{128}$

$5^{127}$

Solution:

Since $3 - 2 = 1,$ multiplying the product by $3 - 2$ does not change it. Then $(3-2)(3+2) = 3^2 - 2^2,$ and multiplying by the next factor $(3^2 + 2^2)$ gives $3^4 - 2^4,$ and so on. Each step doubles the exponent.

After using all seven factors, the product telescopes to $3^{128} - 2^{128}.$

Thus, the correct answer is C.

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