# Square Sum Proof

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

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

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

6

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Problem by BogusBoy

## Solution

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

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

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1 = 1 =

1 + 2 = 3 =

1 + 2 + 3 = 6 =

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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

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

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p

4

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1

2

1

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1

8

1

2

^{n}2

^{n}- 12

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