Sum of first m terms of an A.P. is 0. If a be the first term of the A.P., then the sum of next n terms is : (A) 1 ( ) − − + m a m n m (B) 1 ( ) − − + m a m n n (C) 1 ( ) − − + n a m n n (D) 1 ( ) − − + n a m n m

Asked by snehachauhan0402 | 30th Jul, 2020, 09:50: PM

Expert Answer:

Sum of first m terms of an AP is 0
We have
straight S subscript straight m equals straight m over 2 open square brackets 2 straight a plus open parentheses straight m minus 1 close parentheses straight d close square brackets equals 0
rightwards double arrow straight m over 2 open square brackets 2 straight a plus open parentheses straight m minus 1 close parentheses straight d close square brackets equals 0
rightwards double arrow 2 straight a plus open parentheses straight m minus 1 close parentheses straight d equals 0
rightwards double arrow straight d equals negative fraction numerator 2 straight a over denominator open parentheses straight m minus 1 close parentheses end fraction
Sum space of space next space straight n space terms space is space given space by
straight S subscript straight n equals straight S subscript straight m plus straight n end subscript minus straight S subscript straight m
space space space space equals fraction numerator straight m plus straight n over denominator 2 end fraction open square brackets 2 straight a plus open parentheses straight m plus straight n minus 1 close parentheses straight d close square brackets minus straight m over 2 open square brackets 2 straight a plus open parentheses straight m minus 1 close parentheses straight d close square brackets
space space space space equals am plus an plus fraction numerator open parentheses straight m squared plus straight n squared plus 2 mn minus straight m minus straight n close parentheses straight d over denominator 2 end fraction minus am minus fraction numerator open parentheses straight m squared minus straight m close parentheses straight d over denominator 2 end fraction
space space space space equals an plus straight d over 2 open square brackets straight n squared plus 2 mn minus straight n close square brackets space space equals an minus fraction numerator straight a over denominator open parentheses straight m minus 1 close parentheses end fraction open square brackets straight n squared plus 2 mn minus straight n close square brackets
space space space space equals an minus fraction numerator an over denominator straight m minus 1 end fraction open square brackets straight n plus 2 straight m minus 1 close square brackets space space equals an open square brackets fraction numerator straight m minus 1 minus open parentheses straight n plus 2 straight m minus 1 close parentheses over denominator straight m minus 1 end fraction close square brackets
space space space space equals an open square brackets fraction numerator straight m minus 1 minus straight n minus 2 straight m plus 1 over denominator straight m minus 1 end fraction close square brackets
space space space space equals an open square brackets fraction numerator negative straight m minus straight n over denominator straight m minus 1 end fraction close square brackets
space space space space equals fraction numerator negative an open parentheses straight m plus straight n close parentheses over denominator open parentheses straight m minus 1 close parentheses end fraction

Answered by Renu Varma | 31st Jul, 2020, 11:55: AM