MA001 Study Guide

Unit 11: Introduction to Sequences and Series

11a. write the terms of a sequence given a formula

Write a few terms of a sequence given an explicit formula for the sequence.
Write a few terms of a sequence given a recursive formula for the sequence.

A sequence is an ordered list of numbers; each number in the list is a term of the sequence. The notation $a_{n}$ denotes the $n$ th term of the sequence; that is, the subscript $n$ indicates the position of the term, so $a_{3}$ represents the third term of a sequence.

Listing all the terms of a sequence is often not feasible, particularly if the sequence continues without end. There are typically two ways to express a formula for a sequence. An explicit formula provides a way to find the $n$ th term of the sequence; terms of the sequence can be found by evaluating the formula for $n=1,2,3, \ldots$ or a specific value of $n$ . Evaluating a sequence expressed with an explicit formula is equivalent to evaluating a rule for a function. In fact, a sequence is a function whose domain is the set of positive integers.

Given a sequence defined by the formula $a_{n}=2 \cdot n^{3}+4$ . To find the first five terms of the sequence, evaluate this formula for $n=1,2,3,4,5$ . Thus, the first five terms of the sequence are $6,20,58,132,254$ .

A second way to describe a sequence is with a recursive formula. A recursive formula consists of two parts: the initial term; and a formula to obtain $a_{n}$ from previous terms. For instance, given a sequence described by the following recursive formula.

$\begin{aligned} &a_{1}=12 \\ &a_{n}=5 \cdot a_{n-1}-6 \end{aligned}$

The first term is 12. Then $a_{2}=5 \cdot a_{1}-6=5 \cdot 12-6=54.$ Next, $a_{3}=5 \cdot a_{2}-6=5 \cdot 54-6=264$ . Continuing, $a_{4}=5 \cdot a_{3}-6=5 \cdot 264-6=1314$ and $a_{5}=5 \cdot a_{4}-6=5 \cdot 1314-6=6564$ . So, the first five terms of the sequence are $12,54,264,1314,6564$ . Notice that the first term was given. Each subsequent term is found by multiplying the previous term by 5 and then subtracting 6.

Review the material in Sequences Defined by an Explicit Formula.

11b. write the formula for a sequence given the first few terms, or a recursive relationship

Given the beginning terms of a sequence, write an explicit or recursive formula for the sequence.
Replace a recursive formula for a sequence with an explicit formula for the sequence.
Compare arithmetic and geometric sequences. What are the differences between the formulas for arithmetic and geometric sequences?

At times, the first few terms of a sequence are given but the rule for the sequence is not stated. In order to determine values of the sequence beyond those that are listed, it is helpful to find a formula to describe the sequence. This is the reverse of the skills used in objective 11a.

To find an explicit formula for a sequence, it is important to look for patterns among the terms. If the terms of the sequence are rational numbers, the patterns for the numerator and denominator might be different. Knowing the powers of the first few positive integers, such as 2, 3, 4, or 5, is often useful to find patterns among the terms. The pattern needs to be written in terms of $n$ , where $n$ is the position of the term in the sequence. Be careful! It is important to check any proposed formula for multiple terms in the sequence to ensure that the pattern holds for more than just the first or second terms of the sequence.

For instance, consider the sequence whose first few terms are $2,5,10,17,26,37, \ldots$ To find a rule, start by checking to see if there is a constant difference between the terms or perhaps a constant ratio between the terms. Neither is visible for this sequence. There might not seem to be a pattern. However, notice that the terms are 1 more than a perfect square. That is $2=1^{2}+1,5=2^{2}+1,10=3^{2}+1,17=4^{2}+1,26=5^{2}+1$ , and $37=6^{2}+1$ . So, a formula for the sequence seems to be $a_{n}=n^{2}+1$ .

A small adjustment in the sequence can lead to a major adjustment in the formula. Suppose the previous sequence had been $5, \, 10, \, 17, \, 26, \, 37, \ldots$ ; that is, the sequence has the same terms except the sequence starts at 5 rather than at 2. Each term of the sequence is still 1 more than a perfect square. Now, however, the number being squared does not correspond to the position of the term but to 1 more than the position of the term. So, for this sequence, the explicit formula is $a_{n}=(n+1)^{2}+1$ .

There are two special types of sequences that regularly appear. An arithmetic sequence is a sequence in which the terms differ by a constant amount, called the common difference and generally denoted $d$ . An arithmetic sequence can be described recursively:

$\begin{aligned} &a_{1} \\ &a_{n}=a_{n-1}+d \end{aligned} \text { for } n \geq 2$

This recursive formula gives rise to the following terms of the sequence:

$a_{1}, a_{1}+d, a_{1}+2 d, a_{1}+3 d, a_{1}+4 d, \ldots$

So, an explicit formula for this arithmetic sequence is $a_{n}=a_{1}+(n-1) d$

Given terms of a sequence that appear to differ by a constant amount, a formula can be written either recursively or explicitly using these two relationships. For instance, given the sequence $12, \, 5, \, -2, \, -9, \,-16, \, \ldots$ The difference between successive terms is $-7$ and the first term is 12. So, the sequence can be described with one of the two formulas below:}}+(n-1) d\).

Recursive: $a_{1}=12; a_{n}=a_{n-1}-7$ for $n \geq 2$

Explicit: $a_{n}=12-7(n-1)$

Although either formula can be used to write terms of the sequence, the explicit formula would typically be used to find a specific term. For example, the 100 th term is $a_{100}=12-7(100-1)=-681$ .

A second type of special sequence is a geometric sequence, which is a sequence in which terms differ by a common factor, called the common ratio and generally denoted $r$ . Just as an arithmetic sequence can be written either recursively or explicitly, a geometric sequence can be described either way.

Recursive: $g_{1}; g_{n}=g_{n-1} \cdot r$ for $n \geq 2$

Explicit: $\quad g_{n}=g_{1} \cdot r^{n-1}$

Notice that the two relationships for a geometric sequence are analogous to those for an arithmetic sequence if addition is replaced by multiplication and the common difference is replaced by the common ratio.

Given a sequence that is known to be geometric, with $g_{2}=100$ and $g_{5}=12.5$ . Writing recursive and explicit formulas for this sequence requires finding the first term and the common ratio. The two given terms can be substituted in the explicit formula to obtain two equations.

$100=g_{1} \cdot r \quad$ and $\quad 12.5=g_{1} \cdot r^{4}$

Solving these two equations simultaneously yields $r=\frac{1}{2}$ so that $g_{1}=200$ . Then the sequence can be described as shown below.

Recursive: $g_{1}=200; g_{n}=\frac{1}{2} \cdot g_{n-1}$ for $n \geq 2$

Explicit: $\quad g_{n}=200 \cdot\left(\frac{1}{2}\right)^{n-1}$

Review the material in Sequences Defined by an Explicit Formula, Write the Terms of an Arithmetic Sequence, and Write the Terms of a Geometric Sequence.

11c. apply the formulas for arithmetic and geometric series

Compare the formula for the sum of the first n terms of an arithmetic series to the formula for the first n terms of a geometric series.
Under what conditions can the sum of an infinite geometric series be found? Identify an appropriate formula that can be used to find the sum of an infinite geometric series when it exists.

A series is the sum of the terms of a sequence. Although it is often not possible to find the sum if the sequence is infinite, it is generally possible to find the sum of a specific number of terms. For the special cases of arithmetic and geometric series, that is, the sums of terms in an arithmetic and geometric sequence, respectively, formulas exist to facilitate finding the sum of the first $n$ terms.

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first $n$ terms, denoted $S_{n}$ , is given by $S_{n}=\frac{n\left(a_{1}+a_{n}\right)}{2}$ . To find the sum of an arithmetic series, only three values are needed: the first term; the $n$ th term; and the number of terms to be summed.

In contrast, a geometric series is the sum of the terms of a geometric sequence. The sum of the first $n$ terms, again denoted $S_{n}$ , is given by $S_{n}=\frac{g_{1}\left(1-r^{n}\right)}{1-r}$ for $r \neq 1$ . Notice that to find the sum of a geometric series, again only three values are needed: the first term; the common ratio; and the number of terms to be summed.

Given the sequence: $8, \, 11, \, 14, \, 17, \, 20, \ldots$ . To find the sum of the first 100 terms of the related series, first notice that the terms of the sequence differ by 3, so this would be an arithmetic series. The 100th term is $a_{100}=8+99(2)=206$ . Using the formula for the sum of an arithmetic series yields $S_{100}=\frac{100(8+206)}{2}= 10,700$ .

A sequence that continues indefinitely is an infinite sequence; the sum of the terms of such a sequence is an infinite series. If the infinite series is geometric, then it can be summed under some conditions. In the formula for the sum of a geometric series, the behavior of $r^{n}$ determines whether a sum can be found. If $-1$ , then $r^{n} \rightarrow 0$ as $n \rightarrow \infty$ ; so $S_{n} \rightarrow \frac{g_{1}}{1-r}$ . If $r$ or $r > 1$ , then $r^{n} \rightarrow \infty$ as $n \rightarrow \infty$ , so $S_{n} \rightarrow \infty$ and no sum can be found. For instance, consider the geometric series $81+27+9+3+\ldots$ This is a geometric series with $r=\frac{1}{3}$ ; as the series continues, the terms get smaller and smaller in value. The sum of this infinite geometric series is $S=\frac{81}{1-\frac{1}{3}}=121.5$ .