Question

If ( 1 + x ) ^n = c 0 + c 1 x + c 2 x 2 + ⋯ + c n x n then show that C0+4C1+8C2+12C3+...+4nCn=1+n.2^(n+1)?

Answer 1

See below.

Explanation:

lling

`p(x) = c_0x + 4(1+x)^n` we have

`(dp)/(dx) = c_0+4 n(x+1)^(n-1)=c_0+sum_(k=1)^n4k c_k`

but `c_0=1` then making `x = 1`

`c_0+sum_(k=1)^n4k c_k = 1 + 4n2^(n-1)=1+n2^(n+1)`