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 x 0 is 504 , then x is equal to

Question

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 x > 0 is 504 , then x is equal to, 243, 9, 7, 81

MARKS App · Accepted Answer

Let S = log 9 1 / 2 x + log 9 1 / 3 x + log 9 1 / 4 x + … . . + log 9 1 / 22 x ⇒ S = 2 log 9 x + 3 log 9 x + … … + 22 log 9 x ⇒ S = log 9 x ( 2 + 3 + … . . + 22 ) ⇒ S = log 9 x 21 2 ( 2 + 22 ) ⇒ S = 252 log 9 x Given, S = 504 ⇒ 252 log 9 x = 504 ⇒ log 9 x = 2 ⇒ x = 81

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