If the sum of the first 20 terms of the series _7^1/2x+_7^1/3x+_7^1/4x+ is 460, then x is …

Question

If the sum of the first $20$ terms of the series $\log_{7^{1/2}}x+\log_{7^{1/3}}x+\log_{7^{1/4}}x+\cdots$ is $460$, then $x$ is equal to:

Accepted Answer

460 = log7 x\cdot (2 + 3 + 4 + ..... + 20 + 21)

$$ \Rightarrow $$ 460 = log7 x. $$\left( {{{21 \times 22} \over 2} - 1} \right)$$

$$ \Rightarrow $$ 460 = 230. log7 x

$$ \Rightarrow $$ log7 x = 2

$$ \Rightarrow $$ x = 49

Solutions · 1