Essential Identities/Ideas

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

Practice with These Ideas

How many ordered pairs of integers

(m, n)

satisfy

\sqrt{n^2 - 49} = m

(A) 1(B) 2(C) 3(D) 4(E) infinitely many