1410.12 – A Number Pattern

Find more examples of this pattern:

52−5=42+472−7=62+6\begin{aligned}5^2 - 5 &= 4^2 + 4 \\7^2 - 7 &= 6^2 + 6\end{aligned}

Solution

There are many, many examples. Here is one plus an analysis of why it works:

102−10=92+910(10−1)=9(9+1)10⋅9=9⋅10\begin{aligned}10^2 - 10 &= 9^2 + 9 \\10(10-1) &= 9(9 + 1) \\10 \cdot 9 &= 9 \cdot 10\end{aligned}

In algebra (that is, using formulas) we have

n2−n=n2−2n+1+n−1=(n−1)2+(n−1)\begin{align*}n^2 - n &= n^2 - 2n + 1 + n - 1 \\&= (n-1)^2 + (n-1)\end{align*}

Since this is true for all nn we can create examples at will. For example with n=49n = 49 we find that

492−49=2401−49=2352482+48=2304+48=2352\begin{aligned}49^2 - 49 &= 2401 - 49 = 2352 \\48^2 + 48 &= 2304 + 48 = 2352\end{aligned}

Zowie!