CAT 2021 Slot 3 QA Q20 Logarithms & Exponents

Question 20 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

If \(n\) is a positive integer such that \((\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999\), then the smallest value of \(n\) is

TITA: type in the answer. There are no options and no negative marking.

Answer: 6 Logarithms & Exponents TITA

Solution

  1. The product is \(10^{\frac{1 + 2 + \dots + n}{7}} = 10^{\frac{n(n+1)}{14}}\).
  2. Need \(10^{\frac{n(n+1)}{14}} \gt 999\): \(n = 5\) gives \(10^{30/14} \approx 10^{2.14}\) (too small); \(n = 6\) gives \(10^{42/14} = 1000 \gt 999\).
  3. Smallest \(n = 6\).

Same concept in other years

See every question on Compare powers by summing exponents or taking logs.

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