If the sum of the first m terms of an AP is n and the sum of the first n terms is m,then show that the sum of its first (m+n) terms is-(m+n)

Asked by Samuel Roy | 17th Feb, 2012, 03:57: PM

Expert Answer:

begin mathsize 18px style straight S subscript straight m equals straight m over 2 open parentheses 2 straight a plus open parentheses straight m minus 1 close parentheses straight d close parentheses equals straight n
rightwards double arrow 2 am plus straight m open parentheses straight m minus 1 close parentheses straight d minus 2 straight n equals 0.............. left parenthesis 1 right parenthesis
straight S subscript straight n equals straight n over 2 open parentheses 2 straight a plus open parentheses straight n minus 1 close parentheses straight d close parentheses equals straight m
rightwards double arrow 2 an plus straight n open parentheses straight n minus 1 close parentheses straight d minus 2 straight m equals 0.............. left parenthesis 2 right parenthesis
solve space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis
straight d equals negative fraction numerator 2 open parentheses straight n plus straight m close parentheses over denominator mn end fraction space put space in space left parenthesis 2 right parenthesis
straight a equals fraction numerator open parentheses straight n minus 1 close parentheses open parentheses straight m plus straight n close parentheses minus straight m squared over denominator mn end fraction
straight S subscript straight m plus straight n end subscript equals fraction numerator open parentheses straight m plus straight n close parentheses over denominator 2 end fraction open parentheses 2 open parentheses fraction numerator open parentheses straight n minus 1 close parentheses open parentheses straight m plus straight n close parentheses minus straight m squared over denominator mn end fraction close parentheses plus open parentheses straight m minus 1 close parentheses open parentheses negative fraction numerator 2 open parentheses straight n plus straight m close parentheses over denominator mn end fraction close parentheses close parentheses
simplify space you space will space get space minus open parentheses straight m plus straight n close parentheses
end style

Answered by Arun | 27th Aug, 2018, 02:09: PM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.