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Permutations And Combinations

Question
CBSEENMA11013011

The sums of first n terms of two A.P.’s are in the ratio of 14 – 4n : 3n + 5. Find the ratio of their 8th terms.

Solution

Let straight a subscript 1 comma space space straight a subscript 2 and straight d subscript 1 comma space space straight d subscript 2 respectively be the first term and common difference of two A.P.'s
fraction numerator Sum space of space first space straight n space terms space of space 1 st space straight A. straight P. over denominator Sum space of space first space straight n space terms space of space 2 nd space straight A. straight P. end fraction space equals space fraction numerator 14 minus 4 straight n over denominator 3 straight n plus 5 end fraction  or    WiredFaculty
or        WiredFaculty   or       space space fraction numerator straight a subscript 1 plus open parentheses begin display style fraction numerator straight n minus 1 over denominator 2 end fraction end style close parentheses straight d subscript 1 over denominator straight a subscript 2 plus open parentheses begin display style fraction numerator straight n minus 1 over denominator 2 end fraction end style close parentheses straight d subscript 2 end fraction space equals space fraction numerator 14 minus 4 straight n over denominator 3 straight n plus 5 end fraction
Taking          space fraction numerator straight n minus 1 over denominator 2 end fraction space equals space 7           or         n - 1 = 14                  or         n = 15

∴            fraction numerator straight a subscript 1 plus 7 straight d subscript 1 over denominator straight a subscript 2 plus 7 straight d subscript 2 end fraction space equals space fraction numerator 14 minus 4 left parenthesis 15 right parenthesis over denominator 3 left parenthesis 15 right parenthesis plus 5 end fraction
rightwards double arrow        WiredFaculty

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