If (1 + x + x^2 + x^3)^4n = ∑( r = 0 to 12n) ar. x^r, then find the value of ∑(r = 0 to 3n) a(4r).
(NOTE: In ar, r is a subscript.
In a(4r), 4r is a subscript.)
Answer: 2^(8n - 2)
How to solve this?
Proof
Given we have:
.......
We will show that:
- If we have where thus
- Otherwise, if we can consider the geometric sum:
We have:
Thus, by what we've just proven:
Now set and to get your answer, by the way, if you can write which is very suggestive since we are working with 4th roots of unity