The sums of n terms of three A.P.'s whose fir | vedclass

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(B) Given that the first term $a = 1$ for all three $A.P.'s$ and common differences are $d_1 = 1, d_2 = 2, d_3 = 3$.Using the sum formula $S_n = \frac

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${S_1} + {S_3} = 2{S_2}$

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(B) Given that the first term $a = 1$ for all three $A.P.'s$ and common differences are $d_1 = 1, d_2 = 2, d_3 = 3$.Using the sum formula $S_n = \frac{n}{2}[2a + (n - 1)d]$:$S_1 = \frac{n}{2}[2(1) + (n - 1)1] = \frac{n}{2}[n + 1]$$S_2 = \frac{n}{2}[2(1) + (n - 1)2] = \frac{n}{2}[2n] = n^2$$S_3 = \frac{n}{2}[2(1) + (n - 1)3] = \frac{n}{2}[3n - 1]$Adding $S_1$ and $S_3$:$S_1 + S_3 = \frac{n}{2}[(n + 1) + (3n - 1)] = \frac{n}{2}[4n] = 2n^2$Since $S_2 = n^2$,we have $S_1 + S_3 = 2S_2$.

The sums of $n$ terms of three $A.P.'s$ whose first term is $1$ and common differences are $1, 2, 3$ are ${S_1}, {S_2}, {S_3}$ respectively. The true relation is

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