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# Series Proof

Pro Problems > Math > Logic > Proofs > Proof by Induction

## Series Proof

Use a proof by induction to prove that the first n terms of the series

1
2
+
1
4
+
1
8
+ ... +
1
2n

sums to
2n - 1
2n

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Problem by BogusBoy

## Solution

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## Similar Problems

### Multiples of Pi/4

Prove by induction that for any non-negative integer n,

tan (
(4n + 1)p
4
) = 1

Additional Question: Could this induction be extended to all integers, not just negative ones? If so, how?

Note: Without rigorous proof, we can see that this is true, since the angles which match the equation are all in the first and third quadrant, and all have reference triangles of
p
4
. Nevertheless, this is a good practice exercise in inductive proof.

### Eleven to the N

Prove by induction that for every integer n ≥ 1, 11n is one more than a multiple of ten.

Note: Proof by induction is not the simplest method of proof for this problem, so an alternate solution is provided as well.

### Square Sum Proof

Prove by induction that the sum of the first n positive perfect squares is:

n(n + 1)(2n + 1)
6

### Product and Power

Prove by induction that for all integers n>3:

3n > 9n

### Sum of Integers Proof

1  = 1 =
1(1 + 1)
2

1 + 2 = 3 =
2(2 + 1)
2

1 + 2 + 3 = 6 =
3(3 + 1)
2

1 + 2 + 3 + 4 = 10 =
4(4 + 1)
2

1 + 2 + 3 + 4 + 5 = 15 =
5(5 + 1)
2

It appears from this that the sum of the first n positive integers is
n(n + 1)
2
. Can you prove this by induction?

### Geometric Sum Proof

Give a proof by induction to show that for every non-negative integer n:

20 + 21 + 22 + ... + 2n = 2n + 1 - 1