S_n(x) = log_a^1/2x + log_a^1/3x + log_a^1/6x + log_a^1/11x + log_a^1/18x + log_a^1/27x + ...... up to n-terms, where a > 1. If S₂₄(x) = 1093 and S₁₂(2x) = 265, then value of a is equal to ____________.

${S_n}(x) = {\log _a}{x^2} + {\log _a}{x^3} + {\log _a}{x^6} + {\log _a}{x^{11}}$<br><br>${S_n}(x) = 2{\log _a}x + 3{\log _a}x + 6{\log _a}x + 11{\log _a}x + ......$<br><br>${S_n}(x) = {\log _a}x(2 + 3 + 6 + 11 + .....)$<br><br>${S_r} = 2 + 3 + 6 + 11$ <br><br>$\therefore$ T_n = 2 + (1 + 3 + 5 +......+ (n - 1)) <br><br>= 2 + ${{n - 1} \over 2}\left[ {2.1 + \left( {n - 2} \right)2} \right]$ <br><br>= 2 + $\left( {n - 1} \right)\left[ {1 + \left( {n - 2} \right)} \right]$ <br><br>= n² - 2n + 3 <br><br>General term ${T_r} = {r^2} - 2r + 3$<br><br>${S_n}(x) = \sum\limits_{r = 1}^n {{{\log }_a}x({r^2} - 2r + 3)}$<br><br>${S_{24}}(x) = \sum\limits_{r = 1}^{24} {{{\log }_a}x({r^2} - 2r + 3)}$<br><br>${S_{24}}(x) = {\log _a}x\sum\limits_{r = 1}^{24} {({r^2} - 2r + 3)}$<br><br>$1093 = 4372{\log _a}x$<br><br>${\log _a}x = {1 \over 4}$<br><br>$x = {a^{1/4}}$ .....(i)<br><br>${S_{12}}(2x) = {\log _a}(2x)\sum\limits_{r = 1}^{12} {({r^2} - 2r + 3)}$<br><br>$265 = 530{\log _a}(2x)$<br><br>${\log _a}(2x) = {1 \over 2}$<br><br>$2x = {a^{1/2}}$ ....(ii)<br><br>From (i) and (ii), we get<br><br>$2{a^{{1 \over 4}}} = {a^{{1 \over 2}}}$ <br><br>$\Rightarrow$ ${\left( {2{a^{{1 \over 4}}}} \right)^4} = {\left( {{a^{{1 \over 2}}}} \right)^4}$ <br><br>$\Rightarrow$ 16$a$ = $a$² <br><br>$\Rightarrow$ $a = 16$