Question

i) Write down the first three terms in the expansion of (2-x/2)^n in ascending powers of x, where n is an integer greater than 2.

The first two terms in the expansion of (1+2x)(2-x/2)^n in ascending powers of x are a+bx^2, where a and b are constants.
ii) Find the value of n.
iii) Find the value of a and b.

Explanations if possible, and thank you!

Answer 1

(2 - x/2)^n = 2^n + (nC1)2^(n-1)(-x/2) + (nC2)2^(n-2)(-x/2)^2 + ...
= 2^n - 2^(n-2) n x + 2^(n-5) n(n-1) x^2 - ...

now just multiply that by (1+2x)
2^n - 2^(n-2) n x + 2^(n-5) n(n-1) x^2 ...
+ 2x(2^n - 2^(n-2) n x + 2^(n-5) n(n-1) x^2 ...)
= 2^n + (2^(n+1) - 2^(n-2) n) x ...
Looks like
a = 2^n
b = -2^(n-2) (n-8)