Question

Find the sum of two n terms of the series 1 square minus 2 square + 3 square - 4 square + 5 square minus 6 squared plus 7 squared minus 8

Answer 1

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)

$s_n = \frac{n}{2} (2a + (n - 1)d) \\ \\ \\ = \frac{n}{2} ( - 6 + (n - 1)( - 4)) \\ \\ \\ = \frac{n}{2} ( - 6 - 4n + 4) \\ \\ \\ = \frac{n}{2} ( - 4n - 2) \\ \\ \\ = n( - 2n - 1) = - (2 {n}^{2} + 1)$

Thanks!

Answer 2

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/2[2a + (n-1)d]

Sn=n/2[2*-3+(n-1)*-4]

Sn=n/2[-6+4-4n]

Sn=n/2[-2-4n]

Sn=[-n-2n^2]

Sn= -n[2n + 1]............answer