(a) (a + b)2 + (c + d)2 + 2(b + c)2
a, b, c, d are in G.P.
⇒ b = ar, c = ar2, d = ar3, where r is the common ratio of the G.P.
LHS = (a + b + c + d)2 = (a + ar + ar2 + ar3)2
= {a (1 + r + r2 + r3)}2 = {a ((1 + r) + r2 (1 + r))}2
= {a (1 + r) (1 + r2)}2 = a2 (1 + r)2 (1 + r2)2
= a2 (1 + r)2 (r4 + 2r2 + 1)
= a2 (r4 (1 + r)2 + 2r2(1 + r)2 + (1 + r)2)
= a2r4 (1 + r)2 + 2a2r2(1 + r)2 + a2(1 + r)2
= (ar2 + ar3)2 + 2(ar + ar2)2 + (a + ar)2
= (c + d)2 + 2(b + c)2 + (a + b)2.