Find the sum of 2n terms of the series :1² -2² +3² -4² +5² -6²+.........

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Please find below the solution to the asked query:

We have series  :  1² - 2² + 3² - 4² + 5² - 6² + . . .

We can write our series , As :

$\Rightarrow$( 1² - 2² ) + ( 3² - 4² ) + ( 5² - 6² ) + . . .

$\Rightarrow$[ ( 1 - 2 ) ( 1 + 2 ) ] + [ ( 3 - 4 ) ( 3 + 4 ) ] + [ ( 5 - 6 ) ( 5 + 6 ) ]+ . . .

$\Rightarrow$( -1 ) ( 3 ) + ( -1 ) ( 7 ) + ( -1 ) ( 11 )+ . . .

$\Rightarrow$( - 3 ) + ( - 7 ) + ( - 11 ) + . . .

Here first term  =  a  =  - 3

And

Common difference  = d  = - 7 - ( - 3 ) = - 7 + 3 = - 4 , - 11 - ( - 7 ) = -11 + 7 = - 4

We have formula for sum of n terms in A.P.  , As :

S_n = $\frac{n}{2}\left[2a + \left(n - 1 \right)d\right]$
Here
n  =  Number of terms = 2 n
a  =  first term = - 3
and
d =  Common difference = - 4

So,

S_2n = $\frac{2n}{2}\left[2\left( -3\right) + \left(2n - 1 \right)\left( - 4 \right)\right] = n\left[ - 6 + \left( -8n + 4 \right)\right]= n\left[ - 6 + -8n + 4 \right]= n\left[ -8n - 2 \right] = - 8n^2 - 2n = \mathbf{-}\left(\mathbf{ }\mathbf{ }\mathbf{8}{\mathbf{n}}^{\mathbf{2}}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{2}\mathbf{n}\mathbf{ }\right)\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)}$
Hope this information will clear your doubts about Arithmetic Progressions.

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