
Arithmetic Series
1. Find the equation that gives the sum of the first n positive integer
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the answer given was n(n+1)/2  but I don't know how to get it
2. Show that the sum of the first n odd integers is equal to the perfect square n^2
3. Show that the sum of the first n even integers is equal to n^2 + n

Let $\displaystyle n$ be a positive integer. We will write $\displaystyle S_{n}$ the sum of the first $\displaystyle n$ integers.
1)
$\displaystyle 1 + 2 + ... + n = S_{n}$
$\displaystyle n + (n1) + ... + 1 = S_{n}$
So if we sum these two equations, we get
$\displaystyle (n+1) + ((n1)+2) + ... + (n+1) = 2S_{n}$ , that is to say
$\displaystyle (n+1) + (n+1) + ... + (n+1) = n(n+1) = 2S_{n}$
Therefore $\displaystyle S_{n}=\frac{n(n+1)}{2}$
2)
What are the first $\displaystyle n$ odd integers? $\displaystyle 1,3,...,2n1$
So their sum is
$\displaystyle \sum\limits_{k=1}^{n}(2k1)=\sum\limits_{k=1}^{n}2k\sum\limits_{k=1}^{n}1=2\sum\limits_{k=1}^{n}kn=n(n+1)n=n(n+1n)=n^{2}$
3)
It's easier than 2) :)