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I am having a little trouble following an example I came across today which says that:

$$2 \sum_{k=1}^{n} \sum_{i=0}^{k-2} 1 = 2 \sum_{k=1}^{n} (k-1) = 2 \sum_{j=0}^{n-1} j$$

I have tried fidgeting around with the expressions, but I just don't follow the thought process here. If anyone can please help me with some intermediate steps here, I will be very grateful!

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3 Answers 3

up vote 3 down vote accepted

Let's start by the interior summation, which is $$ \sum_{i=0}^{k-2} 1=\underbrace{1+1+\cdots +1}_{k-1\, \text{times}}, $$ k-1 times because there are the term when $i=0, i=1,\ldots , i=k-1$. So know you have $$ 2\sum_{k=1}^{n}(k-1) $$

Take $j=k-1$. When $k=1,j=0$ and when $k=n,j=n-1$, yielding $$ 2\sum_{j=0}^{n-1}j $$

Note that the $j=0$ term is useless, so you could always take it out, and you can explicitly calculate this sum with this formula: $$ \sum_{j=1}^n j= \frac{n(n+1)}{2}. $$ In your case, one gets $$ 2\sum_{j=0}^{n-1}j=\frac{2(n-1)n}{2}=n(n-1) $$

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Fantastic! Thanks a lot. It seems so easy now :) –  Kristian Oct 2 '12 at 19:55
@Kristian Added some additional info –  Jean-Sébastien Oct 2 '12 at 19:59
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We will take each step one by one
$\sum_{i=0}^{k-2} 1 = k-1$
Now since we are summing up 1 $k-1$ times we can directly replace it with the term $k-1$. Consider the second part.
$\sum_{k=1}^n k-1$
Now put $j = k-1$ in the above equation we get
$\sum_{k=0}^{n-1} j$

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Thank you very much. I really overcomplicated this in my notes! Now it looks so easy I almost feel ashamed for posting here! Really appreciate it! –  Kristian Oct 2 '12 at 19:53
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The first equality: You have $k-1$ summands (one for each $i=0, \ldots k-2$) of $1$, so you can replace the sum by $k-1$. The second equality: An index shift. Instead of adding up $k-1$ for $k=1, \ldots, n$, you add up $j$ for $j=0, \ldots n-1$.

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A more encouraging starting line could be "It's not that difficult". =) –  Pedro Tamaroff Oct 2 '12 at 19:49
Thank you very much! Yes, it seems so obvious now :). Really appreciate your help. –  Kristian Oct 2 '12 at 19:54
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