Question

Suppose a_n = 2ⁿ - 2*a_n-1 and a₀=0

What is a_2n?

Answer 1

Write down a few first terms of a sequence:

$a_0=0\\ a_1=2-0=2\\ a_2=2^2-2\cdot2=0\\ a_3=2^3-0=2^3\\ a_4=2^4-2\cdot2^3=0$

it is easy to see that the even terms of a sequence are zero, so $a_{2n}=0$