Use math induction to prove that each statement is true for every positive integer n?

Question

1/2+1/4+1/8+...+1/2^n=1-1/2^n

Answer 1

n=1: 1/2^1 = 1 - 1/2^1. True for n = 1.

Now we must show that "true for n = k" implies "true for n = k+1". We do this by assuming the premise, and then we must use that to prove the conclusion. So assume:

1/2 + 1/4 + ... + 1/2^k = 1 - 1/2^k

Then we substitute the right hand side into the expression for k+1:

1/2 + 1/4 + ... + 1/2^(k+1) = 1 - 1/2^k + 1/2^(k+1)

But 2^(k+1) can also be written as 2 * 2^k, right hand side becomes:

1 - (2 - 1) / (2 * 2^k) = 1 - 1/2^(k+1)

Putting it together:

1/2 + 1/4 + ... + 1/2^(k+1) = 1 - 1/2^(k+1)

Completing the proof by induction.

Answer 2

prove it is true for 1

(1/2)^1 = 1/2

1-(1/2)^1 = 1/2

now assume it is true for some integer k

1/2 + ... 1/2 ^ k = 1- 1/2^ k

now prove it is true for (k+1)

S( k+1) = 1-(1/2)^k + 1/2^(k+1)

S(k+1) = 1-(1/2)^k+1/2 (1/2^k)

S(k+1)=1- 1/2 (1/2)^k

S(k+1)= 1- (1/2)^ (k+1)

since it is also true for (k+1) this integer must be true for all positive integer n

hope this helps