A) \[\left\{ \frac{r(r+1)}{2x} \right\}+ry\]
B) \[\left\{ \frac{r(r-1)}{2x} \right\}\]
C) \[\left\{ \frac{r(r-1)}{2x} \right\}-ry\]
D) \[\left\{ \frac{r(r+1)}{2y} \right\}-rx\]
Correct Answer: A
Solution :
On putting \[n=1,2,3,.....\] First term of the series\[a=\frac{1}{x}+y\], Second term =\[\frac{2}{x}+y\] \ \[d=\left( \frac{2}{x}+y \right)-\left( \frac{1}{x}+y \right)=\frac{1}{x}\] Sum of \[r\]terms of the series \[=\frac{r}{2}\left[ 2\left( \frac{1}{x}+y \right)+(r-1)\frac{1}{x} \right]\]\[=\frac{r}{2}\left[ \frac{2}{x}+2y+\frac{r}{x}-\frac{1}{x} \right]\] \[=\frac{{{r}^{2}}-r+2r}{2x}+ry=\left[ \frac{r\,(r+1)}{2x}+ry \right]\].
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