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CBSE Class 10 Answered

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 d7016880266 | 20 Jan, 2023, 08:14: PM
answered-by-expert Expert Answer
Let a be the first term and d be the common difference in A.P.

Sum of first m terms of A.P. :-   begin mathsize 14px style S subscript m space equals space m over 2 open parentheses 2 space a space plus space left parenthesis m minus 1 right parenthesis d close parentheses space equals space n space end style ( given )
 
Sum of first n terms of A.P. :-   begin mathsize 14px style S subscript n space equals space n over 2 open parentheses 2 space a space plus space left parenthesis n minus 1 right parenthesis d close parentheses space equals space m space end style ( given )
 
Above expressions can be written as
 
begin mathsize 14px style m space left parenthesis space 2 a space right parenthesis space plus space m space left parenthesis m minus 1 right parenthesis space d space equals space 2 n end style  ..............................(1)
 
begin mathsize 14px style n space left parenthesis space 2 a space right parenthesis space plus space n space left parenthesis n minus 1 right parenthesis space d space equals space 2 m end style  .................... ..........(2)
 
Subtracting eqn.(2) from eqn.(1) , we get
 
begin mathsize 14px style left parenthesis m minus n right parenthesis space left parenthesis 2 a right parenthesis space plus space left parenthesis space m squared minus n squared right parenthesis space d space minus space left parenthesis m minus n right parenthesis d space equals space 2 left parenthesis n minus m right parenthesis end style
 
Dividing above expression by (m-n) , we get
 
begin mathsize 14px style space left parenthesis 2 a right parenthesis space plus space left parenthesis space m plus n right parenthesis space d space minus space d space equals space minus 2 end style
Let us multiply above expression by  ' (m+n)/2 '  and rewrite after simplification as
 
begin mathsize 14px style space fraction numerator left parenthesis m plus n right parenthesis over denominator 2 end fraction space open square brackets 2 a space plus space left parenthesis space m plus n minus 1 right parenthesis space d close square brackets space space equals space minus left parenthesis m plus n right parenthesis end style
LHS of above expression is sum of first (m+n) terms of given A.P.  ( Q.E.D) 

Answered by Thiyagarajan K | 21 Jan, 2023, 08:32: AM

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