How to find an explicit formula for a sequence
When we know the first few terms of a sequence, we can look for a rule that gives the term at any position. Let’s start with a pattern we can see right away, then break a more complicated pattern into smaller pieces.
Match each term to its position
The general term is the term at position . An explicit formula gives directly from . Unless we’re told otherwise, we’ll start with .
In , the term at each position is the negative of that position. So a formula is . Checking , , and gives , , and .
For , the terms rise by each time. The rule gives , which is one short at each position. Adding gives .
Choose the sign for an alternating sequence
At odd positions, . At even positions, . So with indexing from , use for a sequence that starts negative and alternates. Use for a sequence that starts positive and alternates.
Example: Find the sign, numerator, and denominator
Let’s find a formula for a sequence that starts with these four terms.
The first and third terms are negative, so the sign factor is . Ignoring signs, the numerators are , which are the cubes . The denominators are , each more than its position.
Check the sign and size together. At , the formula gives . At it gives . At and it gives and , so all four terms match.
If the magnitudes were squares and the sequence started positive, as in , the same steps would give . For the first two terms, the sign factor gives and , as needed.
Recognize powers, factorials, and other patterns
For , each term is twice the previous term, and , so the pattern gives . In , we multiply by , then , then . That gives .
Checking a proposed formula shows that it reproduces the terms we were given. A finite list can fit more than one rule, so if the exercise gives an extra condition, such as a constant difference, use that condition too.
