Little Difference
What is the value of $2015 \times 2017 - 2016 \times 2016$?
Age
14 to 16
Challenge level
Problem
Without using a calculator, or performing any long multiplication calculations, find the value of:
$2015×2017-2016^2$
What about $2016×2018-2017^2?$
Student Solutions
Answer: $-1$
Let $2016$ or $2017$ be $n$, the calculation is $(n-1)×(n+1) - n^2$
For any $n$, $(n-1)×(n+1) = n^2+n-n-1 = n^2-1$
So $(n-1)×(n+1) - n^2 = n^2-1-n^2=-1$