What does $\sum_{i=1}^{10} 2$ mean exactly?

Question

Suppose I have $$ \large\sum_{i=1}^{10} 2. $$ Do I just add $2$ to itself $10$ times? I have worked on more complex ones with $n$ and such in the place where the $2$ is, but I have never done it when there is just a number there.

Answer 1

You are exactly right. Many summations are of the following form: $$\sum_{i=1}^n c\cdot f(i) = c\cdot \sum_{i=1}^n f(i)$$ Where $f(i)$ is some function of $i$. Notice in this case, $f(i) = 1$, and you are left with $$\sum_{i=1}^{10} 2\cdot f(i) = 2\cdot\sum_{i=1}^{10} 1 = 2\cdot 10$$

Answer 2

The summation of a constant is just the constant time the bound. so in your case the result would be 20. $$\sum_{i=1}^{10} 2 = 2\cdot 10 = 20$$ http://www.psychstat.missouristate.edu/introbook/sbk12m.htm

Answer 3

Read it as $\large\sum_{i=1}^{10} a_i$ where $a_i = 2$ for all $i$.

So it's

$a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 + a_{10} =$

$2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2$

Answer 4

This is another example of how notation can needlessly complicate things. It would be better to write $2 \times 10$ or $10 \times 2$ (too early in the morning for me to argue about commutativity).

If you have any lingering doubts, try this in Wolfram Alpha: Sum[2, {n, 1, 10}].

But don't think that a constant summand in an iterated sum is always pointless. Things like the prime counting function can be defined with a constant summand of 1: $$\pi(n) = \sum_{p \leq n, p \textrm{ prime}} 1.$$