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### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 7: Series basics challenge

# Sum of n squares (part 3)

The sum of the first n squares, 1 + 4 + 9 + 16 + ... + n², is given by the formula ⅙n(n+1)(2n+1). In this video we factor and rewrite the formula that we found in the previous video and obtain the common formula given above. Created by Sal Khan.

## Want to join the conversation?

• If the sum of an arithmetic sequence is an arithmetic series, then would the sum of a geometric sequence be a geometric series?
• Does this sort of thing continue?
Would the sum of cubes end up being divided by 24 with a quartic up top?
Is it always a one degree higher polynomial divided by the original power factorial?
Or am I just hoping it stays this 'easy?'
(1 vote)
• You could write it like that:
(1/24)*6n^4*12n^3*6n^2
but there's a bit of cancelling you could do to make it:
(1/4)n^4+(1/2)n^3+(1/4)n^2
However interestingly it can also be written as:
((1/2)(n)(n+1))^2
which is the square of the sum to n of the natural numbers