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

Dear Student,

Please find below the solution to the asked query:

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

We can write our series , As :

( 12 - 22 ) + ( 32 - 42 ) + ( 52 - 62 ) + . . .

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

( -1 ) ( 3 ) + ( -1 ) ( 7 ) + ( -1 ) ( 11 )+ . . .

( - 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 :

Sn = n22a + n - 1 d
Here
n  =  Number of terms = 2 n
a  =  first term = - 3
and
d =  Common difference = - 4

So,

S2n = 2n22 -3 + 2n - 1  - 4  = n - 6 +  -8n + 4 = n - 6 +   -8n + 4 = n  -8n - 2  = - 8n2 - 2n = -  8n2 + 2n        ( Ans )
Hope this information will clear your doubts about Arithmetic Progressions.

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