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CBSE Class 10 Answered

. If the ratio of sum of π‘š terms of an AP to sum of 𝑛 terms of the same AP is π‘š2 / 𝑛 2 , then prove that the ratio of its π‘šth and 𝑛th terms is 2π‘š − 1 ∢ 2𝑛 − 1.
Asked by yosuraj1234 | 05 Mar, 2019, 03:17: PM
answered-by-expert Expert Answer
Let a and d are first term and common difference of Arithmetic progression
 
begin mathsize 12px style G i v e n space colon space space fraction numerator begin display style m over 2 end style open square brackets 2 a space plus space open parentheses m minus 1 close parentheses d close square brackets over denominator n over 2 open square brackets 2 a space plus space open parentheses n minus 1 close parentheses d close square brackets end fraction space equals space m squared over n squared space space................ left parenthesis 1 right parenthesis

fraction numerator begin display style a space plus fraction numerator open parentheses m minus 1 close parentheses over denominator 2 end fraction space d end style over denominator a space plus fraction numerator open parentheses space n minus 1 close parentheses over denominator 2 end fraction space d end fraction space equals space m over n space

T h e n space space a space plus fraction numerator open parentheses m minus 1 close parentheses over denominator 2 end fraction space d space equals space m cross times k space......... left parenthesis 2 right parenthesis
a space plus fraction numerator open parentheses n minus 1 close parentheses over denominator 2 end fraction space d space space equals space n space cross times space k space space........... left parenthesis 3 right parenthesis

l e t space u s space s u b s t i t u t e space space fraction numerator m minus 1 over denominator 2 end fraction space equals space p minus 1 space space comma space space t h e n space m space equals space 2 p minus 1
u sin g space t h e space a b o v e space s u b s t i t u t i o n space comma space e q n space left parenthesis 2 right parenthesis space b e c o m e s

a space plus space left parenthesis p minus 1 right parenthesis space d space space equals space left parenthesis 2 p minus 1 right parenthesis cross times k space............. left parenthesis 4 right parenthesis

a c c o r d i n g space t o space e q n. left parenthesis 4 right parenthesis comma space space space m minus t h space t e r m comma space space space space space a plus space left parenthesis m minus 1 right parenthesis d space equals space left parenthesis 2 m minus 1 right parenthesis cross times k

n minus t h space t e r m comma space space space space space a plus left parenthesis n minus 1 right parenthesis d space equals space left parenthesis 2 n minus 1 right parenthesis cross times k

h e n c e space r e q u i r e d space r a t i o space comma space space space space space fraction numerator a space plus space left parenthesis m minus 1 right parenthesis d over denominator a space plus space left parenthesis n minus 1 right parenthesis d end fraction space equals fraction numerator 2 m minus 1 over denominator 2 n minus 1 end fraction

end style
Answered by Thiyagarajan K | 05 Mar, 2019, 04:27: PM
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