13.1 Sequences and their notations  (Page 5/15)

Writing the terms of a sequence defined by a recursive formula

Write the first six terms of the sequence defined by the recursive formula.

$$\begin{array}{l}{a}_1=1\\ a_2=2\\ a_n=3a_{n-1}+4a_{n-2}\text{, for }n\ge 3\end{array}$$

The first two terms are given. For each subsequent term, we replace $a_{n-1}$ and $a_{n-2}$ with the values of the two preceding terms.

$$\begin{array}{lllllll}n=3\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & a_3=3a_2+4a_1=3(2)+4(1)=10\hfill \\ n=4\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & a_4=3a_3+4a_2=3(10)+4(2)=38\hfill \\ n=5\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & a_5=3a_4+4a_3=3(38)+4(10)=154\hfill \\ n=6\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & a_6=3a_5+4a_4=3(154)+4(38)=614\hfill \end{array}$$

The first six terms are $\text{{1,2,10,38,154,614}}\text{.}$ See [link] .

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Using factorial notation

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

An example of formula containing a factorial is $a_n=(n+1)!.$ The sixth term of the sequence can be found by substituting 6 for $n.$

The factorial of any whole number $n$ is $n(n-1)!$ We can therefore also think of $5!$ as $5\cdot 4!\text{.}$

Key equations

Key concepts

• A sequence is a list of numbers, called terms, written in a specific order.
• Explicit formulas define each term of a sequence using the position of the term. See [link] , [link] , and [link] .
• An explicit formula for the $n\text{th}$ term of a sequence can be written by analyzing the pattern of several terms. See [link] .
• Recursive formulas define each term of a sequence using previous terms.
• Recursive formulas must state the initial term, or terms, of a sequence.
• A set of terms can be written by using a recursive formula. See [link] and [link] .
• A factorial is a mathematical operation that can be defined recursively.
• The factorial of $n$ is the product of all integers from 1 to $n$ See [link] .