Ratio, Proportion, Indices, LogarithmMCQMTP Sep 24 Series IIQuestion 984 of 305
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If n=m!\displaystyle n = m! where (m\displaystyle 'm' is a positive integer >2\displaystyle > 2) then the value of: 1log2n+1log3n+1log4n++1logmn\displaystyle \frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \frac{1}{\log_4 n} + \dots + \frac{1}{\log_m n}

Options

A1\displaystyle 1
B0\displaystyle 0
C1\displaystyle -1
D2\displaystyle 2
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Correct Answer

Option a1\displaystyle 1

All Options:

  • A1\displaystyle 1
  • B0\displaystyle 0
  • C1\displaystyle -1
  • D2\displaystyle 2

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Detailed Solution & Explanation

Using the property 1logab=logba\displaystyle \frac{1}{\log_a b} = \log_b a:

1log2n+1log3n+1log4n++1logmn\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \frac{1}{\log_4 n} + \cdots + \frac{1}{\log_m n}

=logn2+logn3+logn4++lognm= \log_n 2 + \log_n 3 + \log_n 4 + \cdots + \log_n m

Using the product rule:

=logn(2×3×4××m)= \log_n(2 \times 3 \times 4 \times \cdots \times m)

=logn(m!)= \log_n(m!)

Since n=m!\displaystyle n = m!:

=lognn=1= \log_n n = 1

**The answer is (a) 1.**

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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