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  • If in an A.P., the sum of first m terms is n and the sum of its first n terms is m, then prove that the sum of its first <img alt="(m + n) \: terms\: \: is -(m + n)." src="https://learn.careers360.com/latex-image/?%28m%20&plus;%20n%29%20%5C%3A%20te

If in an A.P., the sum of first m terms is n and the sum of its first n terms is m, then prove that the sum of its first (m + n) \: terms\: \: is -(m + n).

 

 

 

 
 
 
 
 

Answers (1)

\\\text { Given, sum of first m terms of an A.P. is n. }\\ \frac{m}{2}[2 a+(m-1) d]=n\\ 2 a m+m(m-1) d=2n....(i)

\\\text { Given, sum of first n terms of an A.P. is m. }\\ \frac{n}{2}[2 a+(n-1) d]=m\\ 2 a n+n(n-1) d=2m...(ii)

Equation(i) - Equation (ii)

\\2 a(m-n)+\left[\left(m^{2}-n^{2}\right)-(m-n) d\right]=2(n-m) \\ 2 a(m-n)+[(m+n)(m-n)-(m-n) d]=-2(m-n) \\ 2 a+(m+n-1) d=-2 \\ S_{m+n} = \frac{m+n}{2}[2 a+(m+n-1) d]\\ =-2 \times \frac{m+n}{2}=-(m+n)

Hence proved.

 

 

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Safeer PP

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