Question

14. Prove that 2 "c-4"r n

Answer 1

Given expression in the question ∑ r=0 n 3 r C n r prove that it is equal to 4 n .

Binomial Theorem,

∑ r=0 n C n r a n−r b r = ( a+b ) n (1)

Compare both the expression ∑ r=0 n 3 r C n r and ∑ r=0 n C n r a n−r b r ,

Substitute the values of a=1 and b=3 in equation (1)

∑ r=0 n C n r 1 n−r ( 3 ) r = ( 1+3 ) n ∑ r=0 n 3 r C n r = 4 n

Hence proved ∑ r=0 n 3 r C n r = 4 n .