2530.31 – Mystery Linear Operation

Suppose that the operation $\oplus$ is defined so that the following is true:

$\begin{aligned}\oplus 5 &= 11 \\\oplus 8 &= 17 \\\oplus 12 &= 25\end{aligned}$

With this information, find $n$ if $\oplus n = 35$ .

Solution

Note that to get from five to eleven add six. To get from eight to seventeen, add nine. To get from twelve to twenty-five, add thirteen. In each case, the number is added to itself plus one:

$\begin{aligned}\oplus 5 &= 11 &&= 5 + 5 + 1 &&= 2 \times 5 + 1 \\\oplus 8 &= 17 &&= 8 + 8 + 1 &&= 2 \times 8 + 1 \\\oplus 12 &= 25 &&= 12 + 12 + 1 &&= 2 \times 12 + 1\end{aligned}$

Noticing this pattern, we fill in the missing information and find that if $35 = \oplus n = 2 n + 1$ , then $2n = 34$ and $n = 17$ .