If the ratio of the sum of n terms of two A.P | vedclass

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(C) Let $S_n$ and $S'_n$ be the sums of $n$ terms of two $A.P.'s$ with first terms $a, a'$ and common differences $d, d'$ respectively.The ratio of th

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$4:3$

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(C) Let $S_n$ and $S'_n$ be the sums of $n$ terms of two $A.P.'s$ with first terms $a, a'$ and common differences $d, d'$ respectively.The ratio of the sum of $n$ terms is given by:$\frac{S_n}{S'_n} = \frac{\frac{n}{2}[2a + (n - 1)d]}{\frac{n}{2}[2a' + (n - 1)d']} = \frac{7n + 1}{4n + 27}$$\frac{a + \frac{n - 1}{2}d}{a' + \frac{n - 1}{2}d'} = \frac{7n + 1}{4n + 27}$We want to find the ratio of the $11^{th}$ terms,which is $\frac{T_{11}}{T'_{11}} = \frac{a + 10d}{a' + 10d'}$.Comparing $\frac{n - 1}{2}$ with $10$,we get $n - 1 = 20$,so $n = 21$.Substituting $n = 21$ in the ratio:$\frac{T_{11}}{T'_{11}} = \frac{7(21) + 1}{4(21) + 27} = \frac{147 + 1}{84 + 27} = \frac{148}{111} = \frac{4}{3}$.Thus,the ratio is $4:3$.

If the ratio of the sum of $n$ terms of two $A.P.'s$ is $(7n + 1):(4n + 27)$,then the ratio of their $11^{th}$ terms is:

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