I f space S subscript 1 comma S subscript 2 comma S subscript 3 comma......... S subscript m space a r e space t h e space s u m s space o f space n space t e r m s space o f space m space A. P s space w h o s e space f i r s t space t e r m s space a r e space 1 comma 2 comma 3.... m space a n d c o m m o n space d i f f e r e n c e s space a r e space 1 comma 3 comma 5 comma..... left parenthesis 2 m minus 1 right parenthesis space r e s p e c t i v e l y comma space s h o w space t h a t : S subscript 1 space plus space S subscript 2 space plus space S subscript 3 space plus........... S subscript m equals space 1 half m n left parenthesis m n plus 1 right parenthesis

Asked by Olivia Adhikary | 19th Aug, 2014, 10:57: AM

Expert Answer:

F r o m space g i v e n space i n f o r m a t i o n space w e space c a n space w r i t e S equals n over 2 open square brackets 2 a plus open parentheses n minus 1 close parentheses d close square brackets S subscript 1 equals n over 2 open square brackets 2 plus open parentheses n minus 1 close parentheses 1 close square brackets S subscript 2 equals n over 2 open square brackets 4 plus open parentheses n minus 1 close parentheses 3 close square brackets ........ S subscript m equals n over 2 open square brackets 2 m plus open parentheses n minus 1 close parentheses open parentheses 2 m minus 1 close parentheses close square brackets  S u m space o f space a l l space a b o v e space t e r m s equals S subscript 1 plus S subscript 2 plus.... S subscript m equals n over 2 open square brackets 2 fraction numerator m left parenthesis m plus 1 right parenthesis over denominator 2 end fraction plus open parentheses n minus 1 close parentheses m over 2 open parentheses 2 plus open parentheses m minus 1 close parentheses 2 close parentheses close square brackets equals n over 2 open square brackets m open parentheses m plus 1 close parentheses plus open parentheses n minus 1 close parentheses m squared close square brackets equals n over 2 open square brackets m plus m squared n close square brackets equals 1 half open square brackets m n plus m squared n squared close square brackets  H e n c e space P r o v e d

Answered by Dharma Teja | 19th Aug, 2014, 04:21: PM

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