Question

Find the sum of the first 2n terms of the following series. 12 - 22 + 32 - 42
+ .....

Answer 1

We want to find 1² - 2² + 3² - 4² +..... to 2n terms
= 1 - 4 + 9 - 16 + 25 ------------2n terms
= (1 – 4) +(9 – 16)+(25 – 36) + ----------- to n terms. (after grouping)
= -3 +(-7)+(-11)+ -------------- n terms
Now, the above series is in an A.P. with first term a = - 3 and common difference
d = - 4
Now, S_n = n/2[(2a + (n-1)d] = = n/2[(2 x -3) + (n-1)(-4)] = -n(2n + 1).