<s>[INST] How to un-simplify a general term I have a general term of $\frac{3n-1}{5n+7}$ and I want to find out which term $\frac{7}{12}$ is. I have a number in mind that I got from just trial and error but how would I apply a formula to find this. I am treating it as an arithmetic sequence with a common difference of 3 on top and 5 on bottom. I then apply the nth term formula but can't seem to arrive at the right conclusion.
 [/INST] You can't treat $\frac{3n - 1}{5n + 7}$ as an arithmetic progression. For reference, let $a_{i} = \frac{3i - 1}{5i + 7}$. Then,
\begin{align*}
    a_{n} - a_{n - 1} &= \frac{3n - 1}{5n + 7} - \frac{3(n - 1) - 1}{5(n - 1) + 7} \\
    a_{n} - a_{n - 1} &= \frac{3n - 1}{5n + 7} - \frac{3n - 3 - 1}{5n - 5 + 7} \\
    a_{n} - a_{n - 1} &= \frac{3n - 1}{5n + 7} - \frac{3n - 4}{5n + 2} \\
    a_{n} - a_{n - 1} &= \frac{26}{25n^{2} + 45n + 14}
\end{align*}
As you can see, the difference of consecutive terms is not constant.

However, if you are solving for $n$  where $\frac{3n - 1}{5n + 7} = \frac{7}{12}$, just proceed with rules of rational equations.
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