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Sum of N Terms of an AP

In an A.P. th...

Question

In an A.P. the sum of m terms is equal to n and the sum of n terms is m, find the sum of $(m + n)$ terms.

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Solution

$\begin{matrix} S m = n g i v e n S n = m g i v e n S (m t h) = ? \Rightarrow S m = \frac{m}{2} [2 a + (m - 1) d] \Rightarrow (i) 2 m [2 a + (m - 1) d] \Rightarrow s n = \frac{n}{2} [2 a + (n - 1) d] \Rightarrow (i i) 2 m = n [2 a + (n - 1) d] s u b s t r a c t i n g e q u a t i o n (i) f r o m (i i) \Rightarrow 2 a (m - 1) + [(m^{2} - n^{2}) (m - n)] . d = 7 (n - m) \Rightarrow (i i i) 2 a + (m + n - 1) d = 2 ∴ S m t h = [\frac{m t h}{2}] [2 a + (m + n - 1) d] = \frac{(m + n)}{2} \times - 2 = - (m + n) A n s . \end{matrix}$

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If in an AP the sum of m terms is equal to n end the sum of n terms is equal to m then prove that the sum of (m + n) terms is -(m+n)

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