If # (1+x)^n = c_0 + c_1x + c_2x^2 + ... + c_nx^n # then show that # c_0c_1 + c_1c_2 + ... + c_(n-1)c_n = (2n)!(n+1)!(n-1)! #?
1 Answer
Jan 9, 2018
The relationship is invalid
Explanation:
We can readily show that this relationship is invalid by using a counter example:
Consider the case
# (1+x)^2 = 1 + 2x + x^2 #
Then:
# LHS = c_0c_1 + c_1c_2 #
# \ \ \ \ \ \ \ \ = 1.2 + 2.1 #
# \ \ \ \ \ \ \ \ = 2 + 2 #
# \ \ \ \ \ \ \ \ = 4 #
# RHS = (2n)!(n+1)!(n-1)! #
# \ \ \ \ \ \ \ \ = 4!3!1! #
# \ \ \ \ \ \ \ \ = 24.6.1 #
# \ \ \ \ \ \ \ \ = 144 #
Hence the relationship is false.