Find the sum of the following arithmetic progressions: x-y x+y 3x-2y x+y 5x-3y x+y , ..... to n terms.

In an A.P. let first term = a, common difference = d, and there are n terms. Then, sum of n terms is, S_n= n 2 2a + (n - 1)d Given, x-y x+y 3x-2y x+y 5x-3y x+y , ..... Here, First term a= x-y x+y Difference d= 3x-2y x+y - x-y x+y d= 3x-2y-(x-y) x+y d= 3x-2y-x+y x+y d= 2x-y x+y No of terms n = n We know, S_n= n 2 2a + (n - 1)d S_n= n 2 [2 ( x-y x+y +(n-1) (2x-y) (x+y) ] S_n= n 2 [ 2x-2y x+y + n(2x-y)-1(2x-y) x+y ] S_n= n 2 [ 2x-2y+n(2x-y)-2x+y x+y ] S_n= n 2 [ n(2x-y)-y x+y ] S_n= n 2(x+y) [n(2x-y)-y] Hence, sum of n terms is n 2(x+y) [n(2x-y)-y].

Find the sum of the following arithmetic progressions: x-y x+y 3x…

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Question

Find the sum of the following arithmetic progressions:$\frac{\text{x}-\text{y}}{\text{x}+\text{y}}\frac{3\text{x}-2\text{y}}{\text{x}+\text{y}}\frac{5\text{x}-3\text{y}}{\text{x}+\text{y}}, .....\text{ to n terms.}$

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Answer

In an A.P. let first term = a, common difference = d, and there are n terms. Then, sum of n terms is,
$\text{S}_\text{n}=\frac{\text{n}}{2}\{2\text{a} + (\text{n} - 1)\text{d}\}$
Given, $\frac{\text{x}-\text{y}}{\text{x}+\text{y}}\frac{3\text{x}-2\text{y}}{\text{x}+\text{y}}\frac{5\text{x}-3\text{y}}{\text{x}+\text{y}}, .....$
Here,
First term $\text{a}=\frac{\text{x}-\text{y}}{\text{x}+\text{y}}$
Difference $\text{d}=\frac{3\text{x}-2\text{y}}{\text{x}+\text{y}}-\frac{\text{x}-\text{y}}{\text{x}+\text{y}}$
$\Rightarrow\ \text{d}=\frac{3\text{x}-2\text{y}-(\text{x}-\text{y})}{\text{x}+\text{y}}$
$\Rightarrow\ \text{d}=\frac{3\text{x}-2\text{y}-\text{x}+\text{y}}{\text{x}+\text{y}}$
$\Rightarrow\ \text{d}=\frac{2\text{x}-\text{y}}{\text{x}+\text{y}}$
No of terms n = n
We know, $\text{S}_\text{n}=\frac{\text{n}}{2}\{2\text{a} + (\text{n} - 1)\text{d}\}$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}\bigg[2\Big(\frac{\text{x}-\text{y}}{\text{x}+\text{y}}+(\text{n}-1)\frac{(2\text{x}-\text{y})}{(\text{x}+\text{y})}\bigg]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}\bigg[\frac{2\text{x}-2\text{y}}{\text{x}+\text{y}}+\frac{\text{n}(2\text{x}-\text{y})-1(2\text{x}-\text{y})}{\text{x}+\text{y}}\bigg]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}\bigg[\frac{2\text{x}-2\text{y}+\text{n}(2\text{x}-\text{y})-2\text{x}+\text{y}}{\text{x}+\text{y}}\bigg]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2}\bigg[\frac{\text{n}(2\text{x}-\text{y})-\text{y}}{\text{x}+\text{y}}\bigg]$
$\Rightarrow\ \text{S}_\text{n}=\frac{\text{n}}{2(\text{x}+\text{y})}[\text{n}(2\text{x}-\text{y})-\text{y}]$
Hence, sum of n terms is $\frac{\text{n}}{2(\text{x}+\text{y})}[\text{n}(2\text{x}-\text{y})-\text{y}]$.