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The ratio of the sum of n terms of two A.P's is 7n+1:4n+27 . Find the ratio of mth terms.
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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.
Suppose a1, a2 be the first terms and d1, d2 be the common differences of the two given A.Ps.
Then the sums of their n terms are
Concept: Sum of First n Terms of an AP
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Solution : Let `a_(1) "and" a_(2)` be the first terms and `d_(1) "and" d_(2)` be the common difference of the two Aps respectively. <br> Let `S_(n) "and"S'_(n)` be the sum of the first n terms of the two Aps and `T_(n) "and" T'_(n)` be their nth terms respectively. <br> Then, `(S_(n))/(S'_(n)) = (7n + 1)/(4n + 27) rArr ((n)/(2)[2a_(1) + (n-1)d_(1)])/((n)/(2)[2a_(2) + (n-1)d_(2)]) = (7n +1)/(4n +27)` <br> `(2a_(1) + (n-1)d_(1))/(2a_(2) + (n-1)d_(2)) = (7n + 1)/(4n +27). " "... (i)` <br> To find the ratio of mth terms, we replace n by (2m-1) in the above expression. <br> Replacing n by `(2 xx 9-1)`, i.e., 17 on both sides in (i), we get <br> `(2a_(1) + (17-1)d_(1))/(2a_(2) + (17-1)d_(2)) = (7 xx 17 + 1)/(4 xx 17 +27) rArr (2a_(1) +16d_(1))/(2a_(2) + 16d_(2)) = (120)/(95)` <br> `rArr (a_(1) + 8d_(1))/(a_(2) + 8d_(2)) = (24)/(19)` <br> `rArr (a_(1) + (9-1)d_(1))/(a_(2) + (9-1)d_(2)) = (24)/(19)` <br> `rArr (T_(n))/(T'_(n)) = (24)/(19)` <br> `therefore ` required ratio = 24 : 19.