Toolkit

1.
Sum of the first n integers
Proof
$1+2+\cdots+n=\frac{n(n+1)}{2}$
2.
Sum of squares
Proof
$1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$
3.
Sum of cubes
Proof
$1^3+2^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2$
4.
Geometric series
Proof
$1+r+r^2+\cdots+r^n=\frac{r^{n+1}-1}{r-1}$
$|r|<1:\quad 1+r+r^2+\cdots=\frac{1}{1-r}$
5.
Telescoping series
Proof
$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{n(n+1)}=\left(1-\frac{1}{2}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n+1}\right)=1-\frac{1}{n+1}$

Practice with These Ideas

How many ordered pairs of integers

(m, n)

satisfy

\sqrt{n^2 - 49} = m

\text{(A)}\;1

\text{(B)}\;2

\text{(C)}\;3

\text{(D)}\;4

\text{(E)}\;infinitely many