The series \(\frac{1}{1+\log 2}\),\(\frac{1}{1+\log 8}\),and \(\frac{1}{1+\log 32}\) are in

## Answer

Correct Answer : c ) Harmonic progression

Explanation :**Quick Approach**

For such a series if the terms inside the log are in Geometric progression then, the series will be in Harmonic progression.

Hence, (c) is the correct answer.

**Basic Approach**

Consider the given series

\(\frac{1}{1+\log 2}\),\(\frac{1}{1+\log 8}\),and \(\frac{1}{1+\log 32}\)

Here,

Inverse of the numbers of the series will be

\({1+\log 2}\), \({1+\log 8}\), \({1+\log 32}\)

Or,

⇒ \({1+\log 2}\), \({1+\log 2^{3}}\), \({1+\log 2^{5}}\)

⇒ \({1+\log 2}\), \({1+3\log 2}\), \({1+5\log 2}\)

Here,

2^{nd} term – 1^{st} term ⇒ \({1+3\log 2}\)– (\({1+\log 2}\)) = \({2\log 2}\)

Also,

3^{rd} term – 2^{nd} term ⇒\({1+5\log 2}\)– (\({1+3\log 2}\)) = \({2\log 2}\)

As, the common difference (d) of the series is the same

Thus, the inverse of the question series is an arithmetic progression.

And, therefore, the given series is a harmonic series.

Hence, (c) is the correct answer.

Such type of question is asked in various government exams like SSC CGL, SSC MTS, SSC CPO, SSC CHSL, RRB JE, RRB NTPC, RRB GROUP D, RRB OFFICER SCALE-I, IBPS PO, IBPS SO, RRB Office Assistant, IBPS Clerk, RBI Assistant, IBPS RRB OFFICER SCALE 2&3, UPSC CDS etc.

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