Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.

Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.

A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining [latex]{a}_{n}[/latex] in terms of preceding terms. For example, suppose we know the following:

[latex]\begin{align}{a}_{1}&=3 \\ {a}_{n}&=2{a}_{n - 1}-1, \text{for } n\ge 2 \end{align}[/latex]

We can find the subsequent terms of the sequence using the first term.

[latex]\begin{align}{a}_{1}&=3\\ {a}_{2}&=2{a}_{1}-1=2\left(3\right)-1=5\\ {a}_{3}&=2{a}_{2}-1=2\left(5\right)-1=9\\ {a}_{4}&=2{a}_{3}-1=2\left(9\right)-1=17\end{align}[/latex]

So the first four terms of the sequence are [latex]\left\{3,5,9,17\right\}[/latex].

The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms.

[latex]\begin{align}{a}_{1}&=1 \\ {a}_{2}&=1 \\ {a}_{n}&={a}_{n - 1}+{a}_{n - 2}, \text{for } n\ge 3 \end{align}[/latex]

To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We saw above that the eighth and ninth terms are 21 and 34, so

[latex]{a}_{10}={a}_{9}+{a}_{8}=34+21=55[/latex]

Using Factorial Notation

The formulas for some sequences include products of consecutive positive integers. [latex]n[/latex] factorial, written as [latex]n![/latex], is the product of the positive integers from 1 to [latex]n[/latex]. For example,

An example of formula containing a factorial is [latex]{a}_{n}=\left(n+1\right)![/latex]. The sixth term of the sequence can be found by substituting 6 for [latex]n[/latex].

The factorial of any whole number [latex]n[/latex] is [latex]n\left(n - 1\right)![/latex] We can therefore also think of [latex]5![/latex] as [latex]5\cdot 4!\text{.}[/latex]

EXAMPLE: WRITING THE TERMS OF A SEQUENCE USING FACTORIALS

Write the first five terms of the sequence defined by the explicit formula [latex]{a}_{n}=\dfrac{5n}{\left(n+2\right)!}[/latex].

Show Solution

Substitute [latex]n=1,n=2[/latex], and so on in the formula.

[latex]\begin{align}&n=1 && {a}_{1}=\dfrac{5\left(1\right)}{\left(1+2\right)!}=\dfrac{5}{3!}=\dfrac{5}{3\cdot 2\cdot 1}=\dfrac{5}{6} \\[1mm] &n=2 && {a}_{2}=\dfrac{5\left(2\right)}{\left(2+2\right)!}=\dfrac{10}{4!}=\dfrac{10}{4\cdot 3\cdot 2\cdot 1}=\dfrac{5}{12} \\[1mm] &n=3 && {a}_{3}=\dfrac{5\left(3\right)}{\left(3+2\right)!}=\dfrac{15}{5!}=\dfrac{15}{5\cdot 4\cdot 3\cdot 2\cdot 1}=\dfrac{1}{8} \\[1mm] &n=4 && {a}_{4}=\dfrac{5\left(4\right)}{\left(4+2\right)!}=\dfrac{20}{6!}=\dfrac{20}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}=\dfrac{1}{36} \\[1mm] &n=5 && {a}_{5}=\dfrac{5\left(5\right)}{\left(5+2\right)!}=\dfrac{25}{7!}=\dfrac{25}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}=\dfrac{5}{1\text{,}008}\\ \text{ } \end{align}[/latex]

The first five terms are [latex]\left\{\dfrac{5}{6},\dfrac{5}{12},\dfrac{1}{8},\dfrac{1}{36},\dfrac{5}{1,008}\right\}[/latex].

Analysis of the Solution

The figure below shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the factorial term in the denominator results in the denominator becoming much larger than the numerator as [latex]n[/latex] increases. This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero.