The sum of 5 consecutive integers is 10n + 5. What is the median of the 5 integers in terms of n?
How can you prove that it must be 2n + 1?
Call the first of the consecutive integers "i". Then the next four integers are i+1, i+2, i+3, and i+4. The sum of those five numbers is i+ (i+1)+ (i+ 3)+ (i+ 4)+ (i+ 5)= 5i+ 10= 10n+ 5. Solve that for i in terms of n. The "median" is the middle number, i+ 2.
1.
$\displaystyle \underbrace{i+(i+1)+\overbrace{(i+2)}^{median}+(i+ 3)+(i+4)}_{\text{according to HallsofIvy}} = \overbrace{10n+5}^{\text{according to dannyc}}$
$\displaystyle 5i+10=10n+5$
$\displaystyle 5i=10n-5=5(2n-1)$
$\displaystyle \boxed{i=2n-1}$ Now add 2 on both sides of this equation:
$\displaystyle i+\color{red}2 \color{black}= (2n-1)+\color{red}2$
$\displaystyle \boxed{i+2 = 2n+1}$