PRINCIPLE OF MATHEMATICAL INDUCTION=1/2+1/4+1/8+....+1/2n=1-1/2n

Answer 1

test whether it is true for n=1:
1/2 = 1/2^1
yes

Now, assume it is true for n=k. That is,

1/2 + 1/4 + ... + 1/2^k = 1 - 1/2^k
Now, what do we have for n=k+1?
1/2 + 1/4 + ... + 1/2^k + 1/2^(k+1) = 1 - 1/2^k + 1/2^(k+1)
= 1 - 2/2^(k+1) + 1/2^(k+1)
= 1 - 1/2^(k+1)
So, if it's true for n=k, it's true for n=k+1

It's true for n=1, so also for n=2,3,4,...