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Arithmetic Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions3 Marks
Question
Find the sum of the following arithmetic progression:
$\frac{\text{x}-\text{y}}{\text{x}+\text{y}},\ \frac{3\text{x}-2\text{y}}{\text{x}+\text{y}},\ \frac{5\text{x}-3\text{y}}{\text{x}+\text{y}},\ ...$ to n terms.
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Answer

$\frac{\text{x}-\text{y}}{\text{x}+\text{y}},\ \frac{3\text{x}-2\text{y}}{\text{x}+\text{y}},\ \frac{5\text{x}-3\text{y}}{\text{x}+\text{y}},\ ...$ to n terms.
$n^{th}$​​​​​​​ term is above sequence is $\frac{(2\text{n}-1)\text{x}-\text{ny}}{\text{x}+\text{y}}$
Sum of n terms is given by
$\frac{1}{\text{x}+\text{y}}[\text{x}+3\text{x}+5\text{x}+.....+(2\text{n}-1)\\\text{x}-(\text{y}+2\text{y}+3\text{y}...+\text{ny})]$
$=\frac{1}{\text{x}+\text{y}}\Big[\frac{\text{n}}{2}(2\text{x}+(\text{n}-1)2\text{x})-\frac{\text{n}(\text{n}+1)\text{y}}{2}\Big]$
$=\frac{1}{2(\text{x}+\text{y})}[2\text{n}^2\text{x}-2\text{n}^2\text{y}-\text{ny}]$

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