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I know this question is probably really trivial but I really just don't get it and was hoping someone could explain it to me.

With the following two series (where $c$ is a constant):

$$\sum_{i=0}^n\ c = cn +c $$

$$\sum_{i=1}^n\ c = cn $$

I just don't get why they equal what they do. I suppose I'm confused as there is no $i$ term in the expression to which I can substitute actual values into to get the terms, it is just the constant $c$. I just don't know how the terms $cn$ and $cn + c$ came about.

Thank you.

2 Answers 2

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Take the second one to start with. You are just summing ($c$) - the constant value - $n$ times $$ \sum_{i=1}^n\ c = \underbrace{c+c+\ldots + c}_{n\mbox{ times}}=nc\ . $$ In the first case, you are summing $n+1$ times (because the sum starts from $0$, and between $0$ and $n$ you have $n+1$ integers).

It is just the normal thing you would do if the summand depended on the summation index: $$ \sum_{i=1}^n a_i = a_1 + a_2+\ldots + a_n\ . $$ In your case, all the $\{a_i\}$ are identical, and equal to $c$.

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Hint:

$$\underbrace{c+c+c+\dots+c}_n=cn$$

Likewise...

$$c(n+1)=cn+c$$