Let S n ( x ) = log a 1 / 2 x + log a 1 / 3 x + log a 1 / 6 x + log a 1 / 11 x + log a 1 / 18 x + log a 1 / 27 x + … up to n -terms, where a > 1 . If S 24 ( x ) = 1093 and S 12 ( 2 x ) = 265 , then value of a is equal to _____ .

Step-by-step solution

Given S n ( x ) = log a 1 / 2 x + log a 1 / 3 x + log a 1 / 6 x + log a 1 / 11 x + log a 1 / 18 x + log a 1 / 27 x + … + n - terms ⇒ S n ( x ) = ( 2 + 3 + 6 + 11 + 18 + 27 + … + n -terms ) log a x Let S 1 = 2 + 3 + 6 + 11 + 18 + 27 + … + T n S 1 = 2 + 3 + 6 + 11 + 18 + . . . + T n     . . . i S 1 = 0 + 2 + 3 + 6 + 11 + 18 + . . . + T n - 1 + T n     . . . i i Subtract i   from   i i   we get, T n = 2 + 1 + 3 + 5 + . . . + upto  n - 1 -term T n = 2 + (