What is {1}{log₂N}+{1}{log₃N}+{1}{log₄N}+…+{1}{log₁00}N } equal to (N ≠ 1)?

Correct Answer is : 1log100!N{1}{log₁00!}N}log100!N1

Correct Answer is : 1log100!N{1}{log₁00!}N}log100!N1

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UPSC NDA I 2018 Mathematics Solved Paper

Question : 45 of 120

Marks: +1, -0

What is

\frac{1}{log_{2}N}+\frac{1}{log_{3}N}+\frac{1}{log_{4}N}+…+\frac{1}{log_{100}N }

equal to (N ≠ 1)?

Solution:

Concept:

Use the logarithm property

\frac{1}{\log_a b} = \log_b a

to rewrite each term and combine them.

Explanation:

Convert each term of the sum:

\frac{1}{\log_{2}N} = \log_N 2

\frac{1}{\log_{3}N} = \log_N 3

, ...,

\frac{1}{\log_{100}N} = \log_N 100

Add all these logarithms together:

\log_N 2 + \log_N 3 + \cdots + \log_N 100 = \log_N (2 \times 3 \times \cdots \times 100)

The product

2 \times 3 \times \cdots \times 100

equals

100!

(factorial of 100). So the sum becomes

\log_N (100!)

Now apply the same property in reverse:

\log_N (100!) = \frac{1}{\log_{100!} N}

Answer:

\frac{1}{\log_{100!} N}

(option A).

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