Testing whether a value occurs

Set the nth-term expression equal to the proposed value and solve for \(n\). A sequence beginning at \(n=1\) requires a positive integer position; a fractional or nonpositive result does not identify a term.

Example

For \(T_n=3n+1\), test \(85\):

\(\begin{aligned}3n+1&=85\\3n&=84\\n&=28\end{aligned}\)

So \(85\) is the \(28\)th term. For \(86\), the equation gives \(n=\dfrac{85}{3}\), which is not an integer; \(86\) does not occur.