If sum of the first 21 terms of the series log 9 1 / 2 x + log 9 1 / 3 x + log 9 1 / 4 x + … . . where…

If sum of the first 21 terms of the series log91/2x+log91/3x+log91/4x+….. where x>0 is 504, then x is equal to
  1. 243
  2. 9
  3. 7
  4. 81

Solution

Let

S=log91/2x+log91/3x+log91/4x+…..+log91/22x

⇒S=2log9x+3log9x+……+22log9x

⇒S=log9x(2+3+…..+22)

⇒S=log9x212(2+22)

⇒S=252log9x

Given,

S=504

⇒252log9x=504

⇒log9x=2

⇒x=81

Asked in: JEE Main 2021 (20 Jul Shift 2)

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