## Sum of Four Consecutive Integers

How many positive multiples of $10$ that are less than $1000$ are the sum of four consecutive integers?
Source: NCTM Mathematics Teacher

SOLUTION
$1+2+3+4=10$
$2+3+4+5=14$
$3+4+5+6=18$
$4+5+6+7=22$
$5+6+7+8=26$
$6+7+8+9=30$
$7+8+9+10=34$
$8+9+10+11=38$
$9+10+11+12=42$
$10+11+12+13=46$
$11+12+13+14=50$
$\dots$
The pattern is $10,30,50,\dots$ These multiples of $10$ when plotted on a number line are separated by an interval of $20$. How many intervals of $20$ are there from $10$ to $999$?
$999\div 20=49.95$
The number of multiples of $10$ that are the sum of four consecutive integers
$49+1=50$

Answer: $50$