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CBSE Class 10 Answered

Q.If the sum of first m terms of an AP is same as the sum of first n tems (m is not equal to n).Show that the sum of its (m+n) terms is zero.
Asked by tanishqa.rathod | 04 Jan, 2016, 06:05: PM
answered-by-expert Expert Answer

 

L e t space f i r s t space t e r m space o f space A. P. space b e space apostrophe a apostrophe space a n d space c o m m o n space d i f f e r e n c e space b e space apostrophe d apostrophe. S subscript m equals m over 2 open square brackets 2 a plus open parentheses m minus 1 close parentheses d close square brackets S subscript n equals n over 2 open square brackets 2 a plus open parentheses n minus 1 close parentheses d close square brackets m over 2 open square brackets 2 a plus open parentheses m minus 1 close parentheses d close square brackets equals n over 2 open square brackets 2 a plus open parentheses n minus 1 close parentheses d close square brackets rightwards double arrow a m plus d over 2 m open parentheses m minus 1 close parentheses equals a n plus d over 2 n open parentheses n minus 1 close parentheses rightwards double arrow a open parentheses m minus n close parentheses plus d over 2 open square brackets m open parentheses m minus 1 close parentheses minus n open parentheses n minus 1 close parentheses close square brackets equals 0 rightwards double arrow a open parentheses m minus n close parentheses plus d over 2 open square brackets m squared minus n squared minus open parentheses m minus n close parentheses close square brackets equals 0 rightwards double arrow left parenthesis m minus n right parenthesis open square brackets a plus d over 2 open parentheses m plus n minus 1 close parentheses close square brackets equals 0 rightwards double arrow a plus d over 2 open parentheses m plus n minus 1 close parentheses equals 0 space open parentheses because m not equal to n close parentheses rightwards double arrow m plus n minus 1 equals negative 2 a S u m space o f space m plus n space t e r m s space equals S subscript m plus n end subscript equals fraction numerator m plus n over denominator 2 end fraction open square brackets 2 a plus open parentheses m plus n minus 1 close parentheses d close square brackets equals fraction numerator m plus n over denominator 2 end fraction open square brackets 2 a plus open parentheses negative 2 a close parentheses close square brackets equals 0

Answered by | 04 Jan, 2016, 06:54: PM

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