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If the sum of first m terms of an A.P. is n and the sum of first n terms is m, then show that the sum of its first (m + n) terms is

- (m + n).

Asked by Topperlearning User 22nd September 2017, 8:40 AM
Answered by Expert
Answer:

Given, sum of first m terms of an A.P. is n.

Also, sum of first n terms is m.

Subtracting (2) from (1),

begin mathsize 12px style 2 straight a space open parentheses straight m minus straight n close parentheses space plus space open square brackets open parentheses straight m squared space minus space straight n squared close parentheses space minus space open parentheses straight m space minus space straight n close parentheses straight d close square brackets space equals space 2 open parentheses straight n space minus space straight m close parentheses
rightwards double arrow space 2 straight a open parentheses straight m minus straight n close parentheses space plus space open square brackets open parentheses straight m space plus space straight n close parentheses open parentheses straight m space minus space straight n close parentheses space minus space open parentheses straight m space minus space straight n close parentheses straight d close square brackets space equals space minus space 2 open parentheses straight m space minus space straight n close parentheses
rightwards double arrow 2 straight a space plus space open parentheses straight m space plus space straight n space minus space 1 close parentheses space straight d space equals negative 2
rightwards double arrow fraction numerator straight m plus straight n over denominator 2 end fraction open square brackets 2 straight a space plus open parentheses straight m space plus space straight n space minus space 1 close parentheses straight d close square brackets space equals space minus 2 cross times fraction numerator straight m plus straight n over denominator 2 end fraction equals space minus open parentheses straight m space plus space straight n close parentheses end style

 

Hence, the sum of its first (m + n) terms is -(m + n).

Answered by Expert 22nd September 2017, 10:40 AM
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