Sequences and Series

Sequences

Patterns and Rules

A sequence is a list of numbers that follows a pattern. Sequences can be defined using recursive or explicit rules.

A sequence is an ordered list of numbers. Each number in a sequence is called a term of a sequence. The terms in a sequence usually follow a pattern.

  • In the sequence 1, 4, 7, 10, 13, …, each term is 3 less than the next term.
  • In the sequence 3, 6, 12, 24, 48, …, each term is 12\frac{1}{2} of the next term.

The sequence 2, 5, 10, 13, 26 is a finite sequence because it has a limited number of terms. The sequence 20, 10, 5, 2.5, 1.25, … is an infinite sequence because it continues without end. The ellipses (…) at the end of the sequence indicate that it goes on forever.

A recursive rule for a sequence tells how to find any term in the sequence by using the previous term. An explicit rule for a sequence tells how to find any term in the sequence directly, given its position in the sequence. The terms of a sequence are often written as a1,a2,a3,...an,...a_1, a_2, a_3,...a_n,... where the subscript is the position of the term in the sequence.

Arithmetic Sequences

In an arithmetic sequence, consecutive terms differ by the same number, called the common difference.

In an arithmetic sequence, there is a constant difference between consecutive terms. This means the difference between a term and the previous term is always the same value. The difference between consecutive terms in an arithmetic sequence is called the common difference.

The general form for the recursive rule for an arithmetic sequence, where ana_n is the nnth term of the sequence, an1a_{n-1} is the previous term in the sequence, and dd is the common difference is:
an=an1+da_n=a_{n-1}+d
When giving the recursive rule, also include the value of a1a_1, the first term in the sequence. The general form for the explicit rule for an arithmetic sequence where ana_n is the nnth term of the sequence, a1a_{1} is the first term, dd is the common difference, and nn is the position of the term, is:
an=a1+d(n1)a_n=a_1+d(n-1)
Step-By-Step Example
Rules for Arithmetic Sequences
Find a recursive and explicit rule for the sequence 0, 5, 10, 15, 20, 25, 30, …, and find the value of the 21st term in the sequence.
Step 1

Determine the common difference. Subtract each term from the next term.

0, 5, 10, 15, 20, 25, 30, …
50=55-0 = 5
105=510-5=5
1510=515-10=5
Continue until you reach the last given term.

Each term is 5 greater than the previous term. So, the common difference dd is 5.

Step 2
Write the recursive rule. Include the first term and how to find any subsequent terms.
an=an1+da_n=a_{n-1}+d
Substitute d=5d=5.
a1=0an=an1+5\begin{aligned}a_1&=0\\a_n&=a_{n-1}+5\end{aligned}
Step 3
Write the explicit rule.
an=a1+d(n1)a_n=a_1+d(n-1)
Substitute a1=0a_1=0 and d=5d=5.
an=0+5(n1)an=5(n1)=5n5\begin{aligned}a_n&=0+5(n-1)\\a_n&=5(n-1)\\&=5n-5\end{aligned}
Step 4

Use either the recursive rule or explicit rule to find the given term.

The question asks for the 21st term, which would require multiple calculations. It will be more efficient to use the explicit rule.

Substitute n=21n=21.
an=5(n1)a21=5(211)a21=5(20)a21=100\begin{aligned}a_n&=5(n-1)\\a_{21}&=5(21-1)\\a_{21}&= 5(20)\\a_{21}&=100\end{aligned}
Solution
The recursive rule is a1=0a_1=0 and:
an=an1+5a_n=a_{n-1}+5
The explicit rule is:
an=5n5a_n=5n-5
The 21st term in the sequence is 100.

Geometric Sequences

In a geometric sequence, the ratio of consecutive terms is the same number, called the common ratio.

In a geometric sequence, there is a constant ratio between consecutive terms. This means that the ratio between a term and the previous term is always the same value. The ratio between consecutive terms in a geometric sequence is called the common ratio.

The general form for the recursive rule for a geometric sequence, where ana_n is the nnth term of the sequence, rr is the common ratio, and an1a_{n-1} is the previous term in the sequence, is:
an=ran1a_n=r \cdot a_{n-1}
When giving the recursive rule, also include the value of a1a_1, the first term in the sequence. The general form for the explicit rule for a geometric sequence, where ana_n is the nnth term of the sequence, a1a_{1} is the first term, rr is the common ratio, and nn is the position of the term of interest, is:
an=a1rn1a_n=a_1 \cdot r^{n-1}
Step-By-Step Example
Rules for Geometric Sequences
Find a recursive and explicit rule for the sequence 64, 96, 144, 216, … and find the value of the 10th term in the sequence.
Step 1

Identify the common ratio. Divide each term by the previous term:

64, 96, 144, 216, …
96÷64=1.596\div64=1.5
144÷96=1.5144\div96=1.5
216÷144=1.5216\div144=1.5
Each term is 1.5 times as great as the previous term, so the common ratio is 1.5.
Step 2
Write the recursive rule. Include the first term and how to find any subsequent terms.
an=ran1a_n=r \cdot a_{n-1}
Substitute r=1.5r=1.5.
a1=64an=1.5an1\begin{aligned}a_1&=64\\a_n&=1.5a_{n-1}\end{aligned}
Step 3
Write the explicit rule.
an=a1rn1a_n=a_1 \cdot r^{n-1}
Substitute a1=64a_1=64 and r=1.5r=1.5.
an=64(1.5)n1a_n=64(1.5)^{n-1}
Step 4

Use either the recursive rule or explicit rule to find the given term.

The question asks for the 10th term, which would require multiple calculations. It will be more efficient to use the explicit rule.

Substitute n=10n=10.
an=64(1.5)n1a10=64(1.5)101a10=64(1.5)9a10=2,460.375\begin{aligned}a_n&=64(1.5)^{n-1}\\a_{10}&=64(1.5)^{10-1}\\a_{10}&=64(1.5)^9\\a_{10}&=2,\!460.375\end{aligned}
Solution

The recursive rule is a1=64a_1=64 and an=1.5an1a_n=1.5 \cdot a_{n-1}.

The explicit rule is:
an=64(1.5)n1a_n=64(1.5)^{n-1}
The 10th term in the sequence is 2,460.375.

Other Sequences

Some sequences are neither arithmetic nor geometric, such as the Fibonacci sequence, which models many natural phenomena.

Sequences that are neither arithmetic nor geometric may be best described in words as repeating patterns. In the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3 …, the numbers 1, 2, 3 repeat forever. In the sequence 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, …, an additional zero is placed between each pair of ones.

The Fibonacci sequence is a sequence of numbers in which each term is found by adding the two previous terms. The first several terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ….

The Fibonacci sequence is an example of a recursively defined sequence that does not have a common difference or common ratio. The recursive rule for the Fibonacci sequence is a1=1,a2=1,a_1=1, a_2=1, and an=an1+an2a_n=a_{n-1}+a_{n-2}.

Many things in nature grow according to this sequence. The number of leaves on a small plant or petals on a flower often come in numbers that match terms in the Fibonacci sequence.