Multiple choice

The sum upto $(2n+1)$ terms of the series $a^2-(a+d)^2+(a+2d)^2-(a+3d)^2+ ....$ is

  1. $a^2+3nd^2$
  2. $a^2+2nad+n(n-1)d^2$
  3. $a^2+3nad+n(n-1)d^2$
  4. $a^2+2nad+n(2n+1)d^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The series is a^2 - (a+d)^2 + (a+2d)^2 - (a+3d)^2 + ... + (a+2nd)^2. Grouping terms: (a^2 - (a+d)^2) + ((a+2d)^2 - (a+3d)^2) + ... + ((a+(2n-2)d)^2 - (a+(2n-1)d)^2) + (a+2nd)^2. Each pair is of the form x^2 - y^2 = (x-y)(x+y) = -d(2a + (4k+1)d). Summing these leads to the expression a^2 + 2nad + n(2n+1)d^2.

Keep practicing — related questions