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I don't get this formula:

$$\sum_{i=1}^n i^2 = n(n+1)(2n+1)/6$$

Of course, I know that n means that we can plug in its place any natural number and doing the computation. Instead, I am not sure if I get what is the role of i, in this case, as the formula just states that i=1.

In order to understand it, I tried the computation with n = 1 and I have an identity. However, if I try n = 2, on the right side of the equation, should I have 5: 2(2+1) (2*2+1) = 2*3*5 = 30 / 6 = 5... So, I don't understand the formula.

Najib Idrissi
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Always learning
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    The formula means, if you add up all the numbers of the form $i^2$, starting at $i=1$, and going on up to $i=n$, you'll get $n(n+1)(2n+1)/6$. – Gerry Myerson Oct 26 '15 at 08:27
  • But if I try with 2, I have 4 n the left side and 5 on the right side! Where am I wrong? Could you show me it, please? – Always learning Oct 26 '15 at 08:29
  • @NormalHuman, thanks, but I don't know what tag to put according to what the website makes available as tags – Always learning Oct 26 '15 at 08:31
  • If you start with $i=1$, and go on up to $i=2$, then you are adding up the numbers $1^2$ and $2^2$, which gives you 5. Right? – Gerry Myerson Oct 26 '15 at 08:33
  • Do note that $i$ is a mute index here, you could've used another letter here, like: $\sum_{i=1}^n i^2 = \sum_{j=1}^n j^2 = \sum_{k=1}^n k^2 = \cdots$. – Ivo Terek Oct 26 '15 at 08:39

2 Answers2

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The left side is the sum of the squares from $1$ through $n$. If you set $n=2$, the left side is $1^2+2^2=1+4=5$ so the formula is correct.

Ross Millikan
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If you try $n=2$ you get $$1^2+2^2 = \frac{2 \times 3 \times 5}{6}$$ so giving $5$ for the sum of the first two squares, while if you try $n=3$ you get $$1^2+2^2 +3^2= \frac{3 \times 4 \times 7}{6}$$ so giving $14$ for the sum of the first three squares.

You can prove the result by induction.

Henry
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  • yes, this is the exercise actually! Proving it by induction! Just wanted to be sure about the formula before. Now I am! Thank you! – Always learning Oct 26 '15 at 08:33