2022 AMC 12A Problem 8 — Solution | LIVE by Po-Shen Loh

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2022 AMC 12A Problem 8

Below is the professionally curated solution for Problem 8 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.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:exponent geometric sequence

Difficulty rating: 1500

8.

The infinite product

$\sqrt[3]{10}\cdot\sqrt[3]{\sqrt[3]{10}}\cdot\sqrt[3]{\sqrt[3]{\sqrt[3]{10}}}\cdots$

evaluates to a real number. What is that number?

$\sqrt{10}$

$\sqrt[3]{100}$

$\sqrt[4]{1000}$

$10$

$10\sqrt[3]{10}$

Solution:

The $k$ th factor is $10$ raised to the $k$ -fold cube root, namely $10^{1/3^k}.$

The product is $10$ raised to $\frac13+\frac19+\frac1{27}+\cdots=\frac{1/3}{1-1/3}=\frac12.$

So the value is $10^{1/2}=\sqrt{10}.$

Thus, the correct answer is A.

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