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 $n = 17$ .