1410.12 – A Number Pattern

Find more examples of this pattern:

52−5=42+4,72−7=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)n^2 - n = n^2 - 2n + 1 + n - 1 = (n-1)^2 + (n-1)

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!