7.1.2: Explicit Formulas - K12 LibreTexts

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Explicit Formulas

When we represent a sequence with a formula that lets us find any term in the sequence without knowing any other terms, we are representing the sequence explicitly.

Given a recursive definition of an arithmetic or geometric sequence, you can always find an explicit formula, or an equation to represent the nth term of the sequence. Consider for example the sequence of odd numbers we started with: 1, 3, 5, 7, ...

We can find an explicit formula for the nth term of the sequence if we analyze a few terms:

\(\ \begin{array}{l} a_{1}=1 \\ a_{2}=a_{1}+2=1+2=3 \\ a_{3}=a_{2}+2=1+2+2=5 \\ a_{4}=a_{3}+2=1+2+2+2=7 \\ a_{5}=a_{4}+2=1+2+2+2+2=9 \\ a_{6}=a_{5}+2=1+2+2+2+2+2=11 \end{array}\)

Note that every term is made up of a 1, and a set of 2’s. How many 2’s are in each term?

a1 = 1
a2 = 1 + 2 = 3
a3 = 1 + 2 × 2 = 5
a4 = 1 + 3 × 2 = 7
a5 = 1 + 4 × 2 = 9
a6 = 1 + 5 × 2 = 11

The nth term has (n - 1) 2's. For example, a99 = 1 + 98 × 2 = 197 . We can therefore represent the sequence as an = 1 + 2(n - 1). We can simplify this expression:

an = 1 + 2(n - 1)
an = 1 + 2n - 2
an = 2n - 1

In general, we can represent an arithmetic sequence in this way, as long as we know the first term and the common difference, d. Notice that in the previous example, the first term was 1, and the common difference, d, was 2. The nth term is therefore the first term, plus d(n - 1):

an = a1 + d(n - 1)

You can use this general equation to find an explicit formula for any term in an arithmetic sequence.

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