Question

If $S_1,S_2,S_3$ denote the sum of first $11$ terms of three arithmetic progressions whose first terms are unity and their common differences are in harmonic progressions, then the value of $\left(\frac{2S_3{S}_1-S_1{S}_2-S_2{S}_3}{S_1-2S_2+S_3}\right)$ is

Answer 1

Let their common differences be $d_1,d_2,d_3$
Given $d_1,d_2,d_3$ are in H.P.
$\Rightarrow \frac{1}{d_1},\frac{1}{d_2},\frac{1}{d_3}$ are in A.P.
$S_1=\frac{11}{2}[2+(11-1\left)d_1\right]=11(1+5d_1)$
$\Rightarrow d_1=\frac{S_1-11}{55}$

Similarly, $d_2=\frac{S_2-11}{55},d_3=\frac{S_3-11}{55}$
$\Rightarrow \frac{55}{S_1-11},\frac{55}{S_2-11},\frac{55}{S_3-11}$ are in A.P.
$\Rightarrow 55(\frac{1}{S_2-11}-\frac{1}{S_1-11})=55(\frac{1}{S_3-11}-\frac{1}{S_2-11})$
$\Rightarrow \frac{S_1-S_2}{S_1-11}=\frac{S_2-S_3}{S_3-11}$
$\Rightarrow 11(S_1-2S_2+S_3)=2S_3{S}_1-S_1{S}_2-S_2{S}_3$
$\Rightarrow \frac{2S_3{S}_1-S_1{S}_2-S_2{S}_3}{S_1-2S_2+S_3}=11$