If the sum of first n terms of an A.P. is equ | vedclass

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(A) Let the first term be $a$ and the common difference be $d$.The sum of the first $n$ terms is $S_n = \frac{n}{2} \{2a + (n - 1)d\}$.The sum of the

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(A) Let the first term be $a$ and the common difference be $d$.The sum of the first $n$ terms is $S_n = \frac{n}{2} \{2a + (n - 1)d\}$.The sum of the first $m$ terms is $S_m = \frac{m}{2} \{2a + (m - 1)d\}$.Given that $S_n = S_m$,we have:$\frac{n}{2} \{2a + (n - 1)d\} = \frac{m}{2} \{2a + (m - 1)d\}$$n \{2a + (n - 1)d\} = m \{2a + (m - 1)d\}$$2an + n(n - 1)d = 2am + m(m - 1)d$$2a(n - m) + d(n^2 - n - m^2 + m) = 0$$2a(n - m) + d((n^2 - m^2) - (n - m)) = 0$$2a(n - m) + d((n - m)(n + m) - (n - m)) = 0$Since $m 
e n$,we can divide by $(n - m)$:$2a + d(n + m - 1) = 0$Now,the sum of the first $(m + n)$ terms is:$S_{m+n} = \frac{m+n}{2} \{2a + (m + n - 1)d\}$Substituting $2a + (m + n - 1)d = 0$ into the equation:$S_{m+n} = \frac{m+n}{2} 	imes 0 = 0$.

If the sum of first $n$ terms of an $A.P.$ is equal to the sum of its first $m$ terms,$(m \ne n)$,then the sum of its first $(m + n)$ terms will be

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