(i) Let Cr’s denotes the combinatorial coefficients in the expansion of (1 + x)n, n ∈ N. If the integers
an = C0 + C3 + C6 + C9 + ……
bn = C1 + C4 + C7 + C10 + ……
and cn = C2 + C3 + C8 + C11+…..
then prove that
(a) an 3 + bn3 + cn3 – 3anbncn = 2n.
(b) (an – bn)2 + (bn – cn)2 +(cn – an)2=2
(ii) prove the identity:
(C0 – C2 + C4 – C6 + ……… )2+ (C1 – C3 + C5 – C7 + ….. )2 = 2n .
