What is \(\frac{1}{{{{\log }_2}N}} + \frac{1}{{{{\log }_3}N}} + \frac{1}{{{{\log }_4}N}} + \ldots + \frac{1}{{{{\log }_{100}}N\;}}\;\) equal to (N ≠ 1)?

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What is \(\frac{1}{{{{\log }_2}N}} + \frac{1}{{{{\log }_3}N}} + \frac{1}{{{{\log }_4}N}} + \ldots + \frac{1}{{{{\log }_{100}}N\;}}\;\) equal to (N ≠ 1)?

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\(\frac{1}{{{{\log }_{100!}}N}}\)
\(\frac{1}{{{{\log }_{99!}}N}}\)
\(\frac{{99}}{{{{\log }_{100!}}N}}\)
\(\frac{{99}}{{{{\log }_{99!}}N}}\)

Answer 1

Concept:

Formula used:

\({\log _a}b = \frac{1}{{{{\log }_b}a}}\)
Log_a M + log_a N = log_a (MN)

Factorial:

n! = 1 × 2 × 3 × ⋯ × (n – 1) × n

Calculation:

Using \({\log _a}b = \frac{1}{{{{\log }_b}a}}\)

\(\frac{1}{{{{\log }_2}{\rm{N}}}} + \frac{1}{{{{\log }_3}{\rm{N}}}} + \ldots + \frac{1}{{{{\log }_{100}}{\rm{N\;}}}} = {\log _{\rm{N}}}2 + {\log _{\rm{N}}}3 + \ldots + {\log _{\rm{N}}}100\)

= log_N (2 × 3 × ⋯ × 100)

= log_N (100!)

\(= \frac{1}{{{{\log }_{100!}}N}}\)

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What is \(\frac{1}{{{{\log }_2}N}} + \frac{1}{{{{\log }_3}N}} + \frac{1}{{{{\log }_4}N}} + \ldots + \frac{1}{{{{\log }_{100}}N\;}}\;\) equal to (N ≠ 1)?

Answer (Detailed Solution Below)

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