Finding a Term's Value or Its Position

The general-term formula can be used in two directions, depending on what is given and what is asked:

Direction 1 — given the position $n$, find the value $T_n$. Substitute $n$, $a$, and $d$ (or $r$) directly into $T_n=a+(n-1)d$ or $T_n=ar^{n-1}$ and simplify. (This is exactly what §4's two examples did.)

Direction 2 — given the value of some term, find its position $n$. Set the general-term formula equal to the given value and solve the resulting equation for $n$:

• AP: solve $a+(n-1)d = (\text{given value})$ for $n$ — a straightforward linear equation in $n$.
• GP: solve $a\,r^{n-1} = (\text{given value})$ for $n$ — express both sides as a power of the same base where possible and equate exponents, since this syllabus does not require logarithms for this chapter.