Let the terms `a_(1),a_(2),a_(3),…a_(n)` be in G.P. with common ratio r. Let `S_(k)` denote the sum of first k terms of this G.P.. Prove that `S_(m-1)xxS_(m)=(r+1)/r`SigmaSigma_(i le itj le n)a_(i)a_(j)`

Explanation

To prove that \( S_{m-1} \times S_m = \frac{r + 1}{r} \sum_{1 \leq i < j \leq n} a_i a_j \), we will follow these steps:

### Step 1: Define the terms of the G.P.
Let the first term of the G.P. be \( a_1 \) and the common ratio be \( r \). The terms can be expressed as:
\[ a_k = a_1 r^{k-1} \quad \text{for } k = 1, 2, \ldots, n \]

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