If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra (all content)

### Course: Algebra (all content)>Unit 18

Lesson 11: Sum of n squares

# Evaluating series using the formula for the sum of n squares

Using properties of sigma notation to rewrite an elaborate sum as a combination of simpler sums, which we know the formula for. Created by Sal Khan.

## Want to join the conversation?

• Am I just being completely foolish or can you not find the sum of this by finding the area under the curve from 1 to 7?
• This reply is a little late, but hopefully someone will find it useful... You are not being foolish at all! The area under the curve described by the formula given for the series will give the same answer. Calculus and series are related in that way. Sal should give more detail on this in lectures covering Taylor and Maclaurin series. So your question is actually a very good one.
• At Sal talks about looking up the equation on khan academy of the sum of n^2 but where would we look it up at? Is it in the form of a video or a FAQ?
• You can find the explanation for that formula used at in the videos "Finding the sum of n squares part1 and part2"
• @ , Sal gives an example as a sum of fractions. Why? Shouldn't it be 1^2 +2^2+....+7^2?
• He is using a formula to find the sum, rather than finding each value and adding them one by one.
• why doesn't n((a1+a2)/2) work for all of sigma notations?
• The formula n(a1+an)/2 can only be used to find the sum of an arithmetic series with n terms. Notice here that a1 is the first term of the series, and an is the last term. Hence, it cannot be applied to find the sum of squared terms.
• Waait... where does Sal get the 7/6 at @?
• He gets it from the equation at . You can derive it yourself or find derivations for it.
• I don't get what Sal did at . Anybody know?
• At , how did you get both of those formulas for evaluating the sum?
• In it says that 4 goes to 1 to 7 or the sum of 4 from 1 to 7 (n=1)...since n=1 and there is no variable n in it...so why is the answer 7*4=28 or 4+4+4+4+4+4+4=28 ....why isn't the answer just 4 since there is no "n" variable..?
• We have to add 4 to our sum 7 times because we're indexing from n=1 to n=7. The fact that n is not in the summand has no bearing on how we interpret the sigma symbol; it just means that the summand is constant for all indices rather than being a variable term that depends on the index. Hope that makes sense.

Since you seem to be familiar with JavaScript, you might like to think of it this way:
``// This is SUM(n=1 => 7) [ 4 ]var sum = 0;for (var i = 1; i <= 7; i++) {  sum += 4;}// As you can see, we add 4 to the sum 7 times.// ***************// This is SUM(n=1 => 7) [ 4*n ]var sum2 = 0;for (var n = 1; n <= 7; n++) {  sum2 += 4*n;}// The summand depends on n in this sum.``