A series whose n^{th} term is left( frac{n}{x | vedclass

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Question

(A) The $n^{th}$ term of the series is given by $T_n = \frac{n}{x} + y$.To find the sum of $r$ terms,we use the summation formula $S_r = \sum_{n=1}^{r

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$\left\{ \frac{r(r + 1)}{2x} \right\} + ry$

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(A) The $n^{th}$ term of the series is given by $T_n = \frac{n}{x} + y$.To find the sum of $r$ terms,we use the summation formula $S_r = \sum_{n=1}^{r} T_n$.$S_r = \sum_{n=1}^{r} \left( \frac{n}{x} + y \right) = \frac{1}{x} \sum_{n=1}^{r} n + \sum_{n=1}^{r} y$.Using the formula for the sum of the first $r$ natural numbers,$\sum_{n=1}^{r} n = \frac{r(r + 1)}{2}$.Thus,$S_r = \frac{1}{x} \left( \frac{r(r + 1)}{2} \right) + ry$.$S_r = \left\{ \frac{r(r + 1)}{2x} \right\} + ry$.

$A$ series whose $n^{th}$ term is $\left( \frac{n}{x} \right) + y$,the sum of $r$ terms will be

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