ACT® · Math: Functions — Sequences and Series
Explicit Formulas for the nth Term
An explicit formula lets you compute any term of a sequence directly from its position number, without building up through every term before it. For arithmetic sequences that formula is a(n) = a(1) + (n - 1)d.
Say you want the 100th term of a sequence. You could add the common difference 99 times, starting from the first term - slow, and easy to miscount. Or you could notice the pattern: to get to term 100, you take 99 steps of size d away from the first term. That's it. That's the whole idea behind an explicit formula. It trades repetition for arithmetic: one multiplication, one addition, done.
The formula a(n) = a(1) + (n - 1)d packs this logic into symbols. a(1) is where you start. d is the size of each step. And (n - 1) counts how many steps you've taken to reach term n - not n steps, because the first term itself takes zero steps to reach. For 5, 12, 19, 26, the first term is 5 and d = 7, so a(n) = 5 + (n - 1)(7) = 7n - 2. The 15th term is 7(15) - 2 = 103, no counting required. The same formula runs in reverse: for 3, 8, 13, 18, a(n) = 3 + (n - 1)(5) = 5n - 2. Setting 5n - 2 > 200 gives n > 40.4, so the 41st term is the first one past 200.
This formula is the backbone for everything else in the unit. Once you can name any term directly, you can also sum long stretches of a sequence without adding term by term, and you can compare arithmetic patterns to geometric ones, where a fixed ratio replaces the fixed difference. Watch how the formula shifts if a problem numbers its terms starting at 0 instead of 1 - that's the one place students lose easy points.
Key terms
- Explicit formula
- A rule that computes any term directly from its position number, without needing the terms before it.
- Common difference (d)
- The fixed amount added to each term to get the next one in an arithmetic sequence.
- Recursive rule
- A rule that defines each term using the term immediately before it, rather than its position.
- Indexing
- The choice of which number labels the first term of a sequence, usually 1 or 0.
Watch it explained
- How it worksHow does a(n) = a(1) + (n - 1)d pack an entire sequence into one line?
- Worked exampleFind an explicit formula for 5, 12, 19, 26 and use it to get the 15th term.
- Worked exampleFind which term of 3, 8, 13, 18 is the first one bigger than 200.
- Going furtherWhat happens to that formula when the sequence is indexed from n = 0 rather than n = 1?
Check yourself
End of topic.