Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2021 · Mathematics · Basics of Mathematics

If n=100 !, then what is the value of the following? (1)/( _2...

If n=100n=100 !, then what is the value of the following? 1log⁡2n+1log⁡3n+1log⁡4n+…+1log⁡100n\frac{1}{\log _2 n}+\frac{1}{\log _3 n}+\frac{1}{\log _4 n}+\ldots+\frac{1}{\log _{100} n}

Options

  1. A00
  2. B11
  3. C22
  4. D33

Correct answer

B. 11

Step-by-step solution

From \frac{1}{\log _a b}=\log _b a\frac{1}{\log _2 n}+\frac{1}{\log _3 n}+\frac{1}{\log _4 n}+\frac{1}{\log _5 n}+\ldots.+\frac{1}{\log _{100} n}=log⁡n2+log⁡n3+log⁡n4+…..+log⁡n100=log⁡n(2⋅3⋅4⋅5⋅…..100)=log⁡n100!As n=100! (given) ∴log⁡100!100!=1.\begin{aligned} & =\log _n 2+\log _n 3+\log _n 4+\ldots..+\log _n 100 \\ & =\log _n(2 \cdot 3 \cdot 4 \cdot 5 \cdot \ldots..100)=\log _n 100! \\ & \text{As } n=100!\quad \text{ (given) } \\ & \therefore \log _{100!} 100!=1.\end{aligned}
Practice Basics of Mathematics on Quantrex Academy →

More from Basics of Mathematics

All Basics of Mathematics questions Full Basics of Mathematics list Basics of Mathematics JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs