# Geometric Sum Proof

Pro Problems > Math > Logic > Proofs > Proof by Induction## Geometric Sum Proof

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

2^{0} + 2^{1} + 2^{2} + ... + 2^{n} = 2^{n + 1} - 1

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Problem by BogusBoy

## Solution

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Click here to assign this problem to your students.## Similar Problems

### Eleven to the N

Prove by induction that for every integer n ≥ 1, 11^{n} 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.*

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

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tan ((4n + 1)p

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p

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### Sum of Integers Proof

1 = 1 =

1 + 2 = 3 =

1 + 2 + 3 = 6 =

1 + 2 + 3 + 4 = 10 =

1 + 2 + 3 + 4 + 5 = 15 =
It appears from this that the sum of the first n positive integers is . Can you prove this by induction?

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

n(n + 1)

2

### Series Proof

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

1

2

1

4

1

8

1

2

^{n}2

^{n}- 12

^{n}### Product and Power

Prove by induction that for all integers n>3:

3^{n} > 9n

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