Question

In an AP of three consecutive terms (x-y)² , (x²+y²) , (x+y)², the common difference is ​

Answer 1

$\red{\frak{hello}}$

In an AP whose terms are

$\tt \: a_1, \: a_2,\: a_3, \: a_4......a_n\\ \\ \tt \: common \: difference \: = a2 - a1 = a3 - a2$

here

terms of AP are

$\tt{(x - y)}^{2} ,\: ( {x}^{2} + {y}^{2} ) ,\: {(x + y)}^{2}$

So, common difference (d) will be

$\tt \: d = {x}^{2} + {y}^{2} - {(x - y)}^{2} \\ \\ \tt \: d = {x}^{2} + {y}^{2} - ( {x}^{2} + {y}^{2} - 2xy) \\ \\ \tt \: d = {x}^{2} + {y}^{2} - {x}^{2} - {y}^{2} + 2xy \\ \\ \tt \boxed{ d = 2xy}$

$\pink{\frak{common \: difference \: of \: ap \: will \: be \:\tt \: 2xy}}$

♣︎Verifying the answer♣︎

adding 2xy to the first term we should get second term

so,

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

= x^2 + y^2

that is the second term.

also,

on adding 2xy to the second term we should get third term

so,

x^2+y^2 +2xy = x^2 + y^2 +2xy

= (x+y) ^2

that is the third term.

hence,

verified that

2xy is the common difference of the given AP.

Answer 2

Answer:

In an AP, common difference=$a_{k+1}-a_{k}$

So the required common difference is 2xy