Question 71

The value of the expression: $$\sum_{i=2}^{100}\frac{1}{log_{i}100!}$$ is:

Solution

Expression : $$\sum_{i = 2}^{100} \frac{1}{log_{i}100!}$$

= $$\frac{1}{log_2 100!} + \frac{1}{log_3 100!} + \frac{1}{log_4 100!} +$$ ..... $$+ \frac{1}{log_{100} 100!}$$

We know that $$\frac{1}{log_b a} = log_a b$$

= $$log_{100!} 2 + log_{100!} 3 + log_{100!} 4 +$$ ..... $$+ log_{100!} 100$$

Also, $$log_a b + log_a c = log_a (b \times c)$$

= $$log_{100!} (2 \times 3 \times 4 \times 5 \times ..... \times 100)$$

= $$log_{100!} 100! = 1$$

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