Question

1²-2²+3²-4²+.......2n terms​

Answer 1

Analysing the given series,we can see that

-2^2+1^2=-3.......i)

-4^2+3^2=-7....ii)

This shows that,the differences of two consecutive pairs are in AP.......

Since,we are taking pair as one term in AP....so,total terms in AP,will be 2n/2=n

-3,-7,-11........nterms

Now,

using a+(n-1)d formula,we will find out nth term.....

In that AP,A=-3,

common difference,d=-4

nth term or last term(l)=-3+(n-1)(-4)

=-3-4n+4=1-4n

now,

we know

sum of AP=n/2 ×(a+l)

=n/2×(-4+1-4n)

=n/2 ×(-3-4n)=>-3n/2 -4n^2/2=(-3n-4n^2)/2

so,sum of the given series=-(3n+4n^2)/2

{hope it helps}