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, 20:14: PM
Let a be the first term and d be the common difference in A.P.
Sum of first m terms of A.P. :-
( given )
![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](https://images.topperlearning.com/topper/tinymce/cache/d842207a3aa5bb4292c8608e031fb634.png)
Sum of first n terms of A.P. :-
( given )
![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](https://images.topperlearning.com/topper/tinymce/cache/e6ca298568721b940997b36ce1343a71.png)
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](https://images.topperlearning.com/topper/tinymce/cache/bc3c21e62b0f3cabd5e4433c6ca3281d.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/f389dca728488d9fe0c6b327be1a31cb.png)
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](https://images.topperlearning.com/topper/tinymce/cache/5c9ea28cc47e9142e9bc22cf3614217e.png)
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](https://images.topperlearning.com/topper/tinymce/cache/14cbc97ac9fa0722d898c4f76ac64b0c.png)
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](https://images.topperlearning.com/topper/tinymce/cache/c11f33b03f7682a97ec8437ae913b4fa.png)
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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