# How should I solve this summation problem?

Lets say that we have these $x$ and $y$ coordinates $x=1,2,3,4,5$ and $y=6,7,8,9,10$ and where $n=5$. How would I use these $x$ coordinates with the first summation? Now, I know that learning is important and it's never a good idea to get other people to do the question for you because you'll never learn so I am hoping that someone can please help me with the first summation. The rest I can do myself after someone explains the first one.

Since my $n=5$ and I have $x=1,2,3,4,5$ does that mean that I have to do the summation of all five individual $x$'s then add all $5$ summations of each individual $x$?

Find the five sums:

$$\sum_{i=1}^n x_i, \quad \sum_{i=1}^n y_i, \quad \sum_{i=1}^n x_i^2, \quad \sum_{i=1}^n y_i^2, \quad \sum_{i=1}^n x_i y_i.$$

$$\sum_{i=1}^5 = x_1 + x_2 + x_3 +x_4 + x_5 = 1+2+3+4+5 = 15$$

$\textbf{Additional}$: Even though you didn't specify the order of the $x_i 's$, since we have associativity everything is fine.

• So I don't do the summation of each of my x's...? – Hard Worker Jun 7 '14 at 3:12
• I'm not understanding what you are saying. This sums all $x_i$. – Mr.Fry Jun 7 '14 at 3:15
• What I meant was that I thought I was supposed to do the summation of each individual x – Hard Worker Jun 7 '14 at 3:17
• Oh ok now I get you. Yeah notice that the index is changing so the value of $x$ will change. – Mr.Fry Jun 7 '14 at 3:18
• By the way, How do you guys do all these crazy summation symbols on keyboard? I tried finding out how to do it, but ended up just using a pic.. – Hard Worker Jun 7 '14 at 3:35

The coordinates you have come in pairs: \begin{align*} (x_1, y_1) &= (1,6) \\ (x_2, y_2) &= (2,7) \\ (x_3, y_3) &= (3,8) \\ (x_4, y_4) &= (4,9) \\ (x_5, y_5) &= (5,10). \end{align*} Therefore, $x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4, x_5 = 5$, and you have $n = 5$ such pairs; so when we write $$\sum_{i=1}^n x_i = x_1 + x_2 + \cdots + x_n,$$ we mean $$1 + 2 + 3 + 4 + 5.$$ Note the fact that these coordinates come in pairs is important, because only then will you know which ones match up when you want to evaluate an expression containing both $x_i$ and $y_i$; e.g., $$\sum_{i=1}^n x_i y_i = (1 \times 6) + (2 \times 7) + (3 \times 8) + (4 \times 9) + (5 \times 10).$$

• Ah..., thank you! I was under the impression that we were supposed to do the summation of each x where I would be doing the summation of 1, then 2 and so on..I am really over thinking this.. – Hard Worker Jun 7 '14 at 3:16