The sums of `n` terms of three arithmetical progressions are `S_1, S_2`and `S_3dot` The first term of each unity and the common differences are `1,2` and `3` respectively. Prove that `S_1+S_3=2S_2dot`

Explanation

To prove that \( S_1 + S_3 = 2S_2 \) for the given arithmetic progressions, let's start by calculating the sums of the first \( n \) terms of each progression.

### Step 1: Define the first terms and common differences
We have three arithmetic progressions (APs):
- For \( S_1 \): First term \( A_1 = 1 \), common difference \( D_1 = 1 \)
- For \( S_2 \): First term \( A_2 = 1 \), common difference \( D_2 = 2 \)
- For \( S_3 \): First term \( A_3 = 1 \), common difference \( D_3 = 3 \)

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