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27. If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.
Asked by ayushshandilya83 | 22 Feb, 2018, 09:44: AM
answered-by-expert Expert Answer
begin mathsize 16px style For space 1 st space straight A. straight P. comma first space term equals straight a subscript 1 comma space common space difference equals straight d subscript 1 comma space sum equals straight S subscript 1 For space 2 nd space straight A. straight P. comma first space term equals straight a subscript 2 comma space common space difference equals straight d subscript 2 comma space Sum equals straight S subscript 2 straight S subscript 1 over straight S subscript 2 equals fraction numerator begin display style straight n over 2 end style left square bracket 2 straight a subscript 1 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 1 right square bracket over denominator straight n over 2 left square bracket 2 straight a subscript 2 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 2 right square bracket end fraction equals fraction numerator 2 straight a subscript 1 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 1 over denominator 2 straight a subscript 2 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 2 end fraction rightwards double arrow fraction numerator 2 straight a subscript 1 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 1 over denominator 2 straight a subscript 2 plus left parenthesis straight n minus 1 right parenthesis straight d subscript 2 end fraction equals fraction numerator 7 straight n plus 1 over denominator 4 straight n plus 27 end fraction To space find space ratio space of space 9 th space terms comma space we space replace space straight n space by space 17 space straight i. straight e. space left parenthesis 2 cross times 9 minus 1 right parenthesis fraction numerator 2 straight a subscript 1 plus left parenthesis 17 minus 1 right parenthesis straight d subscript 1 over denominator 2 straight a subscript 2 plus left parenthesis 17 minus 1 right parenthesis straight d subscript 2 end fraction equals fraction numerator 7 cross times 17 plus 1 over denominator 4 cross times 17 plus 27 end fraction rightwards double arrow fraction numerator 2 straight a subscript 1 plus 16 straight d subscript 1 over denominator 2 straight a subscript 2 plus 16 straight d subscript 2 end fraction equals 120 over 95 rightwards double arrow fraction numerator straight a subscript 1 plus 8 straight d subscript 1 over denominator straight a subscript 2 plus 8 straight d subscript 2 end fraction equals 24 over 19 Thus comma space ratio space of space 9 th space terms equals 24 space colon space 19 end style
Answered by Rashmi Khot | 22 Feb, 2018, 11:57: AM

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