If `S_1,S_2, S_3, S_m` are the sums of `n` terms of `m` A.P. `' s` whose first terms are `1,2,3, ,m` and common differences are `1,3,5, ,(2m-1)` respectively. Show that `S_1+S_2,+S_m=(m n)/2(m n+1)`

Explanation

To solve the problem, we need to find the sums \( S_1, S_2, S_3, \ldots, S_m \) of the first \( n \) terms of \( m \) arithmetic progressions (APs) and show that

\[ S_1 + S_2 + S_3 + \ldots + S_m = \frac{mn}{2}(mn + 1). \]

### Step 1: Define the first terms and common differences of the APs

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