# Sum of Integers Proof

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

Presentation mode

Problem by BogusBoy

## Solution

In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. If you would like to view the solutions to these problems, you must have a Virtual Classroom subscription.Assign this problem

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.*

### 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:

3^{n} > 9n

### 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}### Multiples of Pi/4

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

tan ((4n + 1)p

4

*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. Nevertheless, this is a good practice exercise in inductive proof.*

p

4

### 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

# Featured Games on This Site

Match color, font, and letter in this strategy game

Mastermind variation, with words

# Blogs on This Site

Reviews and book lists - books we love!

The site administrator fields questions from visitors.

Like us on Facebook to get updates about new resources