Question

if a.b.c.d are four consecutive terms of an arithmetic progression then show that a^2 -d^2=3(b^2 - c^2)

Answer 1

the terms are

a, a+d, a+2d, a+3d

so you want to show that

a^2 - (a+3d)^2 = 3((a+d)^2 - (a+2d)^2)

Now just verify that by expanding both sides.