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Find the Sum of the Following Arithmetic Progressions (X - Y)^2,(X^2 + Y^2), (X + Y)^2,.... to N Term - Mathematics

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प्रश्न

Find the sum of the following arithmetic progressions 

`(x - y)^2,(x^2 + y^2), (x + y)^2,.... to n term`

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उत्तर

`(x - y)^2,(x^2 + y^2), (x + y)^2,.... to n term`

Common difference of the A.P. (d) = `a_2 - a_1`

`= (x^2  + y^2) - (x - y)^2`

`= x^2 + y^2 - (x^2 + y^2 - 2xy)`

`= x^2 + y^2 - x^2 - y^2 + 2xy`

= 2xy

Number of terms (n) = n

First term for the given A.P. `(a) = (x - y)^2`

So, using the formula we get,

`S_n = n/2 [2(x - y)^2 + (n -1)2xy]`

Now, taking 2 common from both the terms inside the bracket we get,

`= (n/2)[(2)(x -y)^2 +(2)(n -1)xy]`

`= (n/2)(2)[(x - y)^2 + (n -1)xy]`

`= (n)[(x - y)^2 + (n -1)xy]`

Therefore, the sum of first n terms for the given A.P. is `n[(x - y)^2 + (n -1)xy]`

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पाठ 5: Arithmetic Progression - Exercise 5.6 [पृष्ठ ३०]

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आरडी शर्मा Mathematics [English] Class 10
पाठ 5 Arithmetic Progression
Exercise 5.6 | Q 1.6 | पृष्ठ ३०

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