Question

Find the sum of the first two n terms of the series one square minus 2 square + 3 square minus 4 square + so on

Answer 1

Answer:

Step-by-step explanation:

The series is :-

1² - 2² + 3² - 4² +5² - 6² +7² - 8². . . .

Now,

We know that

A² - B² =(A+B) (A-B)

So,

1² - 2² =(1-2)(1+2) =3(-1) =(-3)

Now ,

In each such numbers - 1 will be constant as they are consecutive numbers.

Also,

If we consider 1² - 2² as 1st term of AP and 3² - 4² as second term and so on...

The AP will be as follows :-

(-3),(-7),(-11),(-15).....

Now ,

To understand this :-

(-3) has two numbers of the series...

Means if we have the AP (-3),(-7), (-11)... And we find the Sum of n terms of this AP we will get the sum of 2n terms of the series given above.

As each term of the AP has 2 terms of the series.

Now,

IN THE AP :-

A= (-3)

D = (-4)

Sn =  n    [ 2A + (n-1) D]

          2

     =   n [ 2(-3) + (n-1)(-4)]

           2

     =     n [ -6 -4n +4]

           2

      =    n [  -4n -2]

            2

      =    n [ 2 (-2n -1) ]

            2

      =    n(-2n-1)

      =  -2n² - n

      = - ( 2n²+n )