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Finding the nth Term Given the Common Ratio and the First Term

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Geometric Sequences and Finding the nth Term Given the Common Ratio and the First Term

The following sequence shows the distance (in centimeters) a pendulum travels with each successive swing. Write a general rule for the geometric sequence.

80, 72, 64.8, 58.32, ...

Guidance

A geometric sequence is a sequence in which the ratio between any two consecutive terms, \frac{a_n}{a_{n-1}} , is constant. This constant value is called the common ratio . Another way to think of this is that each term is multiplied by the same value, the common ratio, to get the next term.

Example A

Consider the sequence 2, 6, 18, 54, \ldots

Is this sequence geometric? If so, what is the common difference?

Solution: If we look at each pair of successive terms and evaluate the ratios, we get \frac{6}{2}=\frac{18}{6}=\frac{54}{18}=3 which indicates that the sequence is geometric and that the common ratio is 3.

More Guidance

Now let’s see if we can develop a general rule ( n^{th} term) for this sequence. Since we know that each term is multiplied by 3 to get the next term, let’s rewrite each term as a product and see if there is a pattern.

a_1 &= 2 \\a_2 &= a_1(3)=2(3)=2(3)^1 \\a_3 &= a_2(3)=2(3)(3)=2(3)^2 \\a_4 &= a_3(3)=2(3)(3)(3)=2(3)^3

This illustrates that the general rule is a_n=a_1(r)^{n-1} , where r is the common ratio. This even works for the first term since a_1=2(3)^0=2(1)=2 .

Example B

Write a general rule for the geometric sequence 64, 32, 16, 8, \ldots

Solution: From the general rule above we can see that we need to know two things: the first term and the common ratio to write the general rule. The first term is 64 and we can find the common ratio by dividing a pair of successive terms, \frac{32}{64}=\frac{1}{2} . The n^{th} term rule is thus a_n=64 \left(\frac{1}{2}\right)^{n-1} .

Example C

Find the n^{th} term rule for the sequence 81, 54, 36, 24, \ldots and hence find the 12^{th} term.

Solution: The first term here is 81 and the common ratio, r , is \frac{54}{81}=\frac{2}{3} . The n^{th} term rule is a_n=81 \left(\frac{2}{3}\right)^{n-1} . Now we can find the 12^{th} term a_{12}=81 \left(\frac{2}{3}\right)^{12-1}=81 \left(\frac{2}{3}\right)^{11}=\frac{2048}{2187} . Use the graphing calculator for the last step and MATH > Frac your answer to get the fraction. We could also use the calculator and the general rule to generate terms seq(81(2/3)^\land(x-1), x, 12, 12) . Reminder: the seq ( \ ) function can be found in the LIST ( 2^{nd} STAT ) Menu under OPS . Be careful to make sure that the entire exponent is enclosed in parenthesis.

Intro Problem Revisit We need to know two things, the first term and the common ratio, to write the general rule. The first term is 80 and we can find the common ratio by dividing a pair of successive terms, \frac{72}{80}=\frac{9}{10} . The n^{th} term rule is thus a_n=80 \left(\frac{9}{10}\right)^{n-1} .

Guided Practice

1. Identify which of the following are geometric sequences. If the sequence is geometric, find the common ratio.

a. 5, 10, 15, 20, \ldots

b. 1, 2, 4, 8, \ldots

c. 243, 49, 7, 1, \ldots

2. Find the general rule and the 20^{th} term for the sequence 3, 6, 12, 24, \ldots

3. Find the n^{th} term rule and list terms 5 thru 11 using your calculator for the sequence -1024, 768, -432, -324, \ldots

4. Find the value of a 10 year old car if the purchase price was $22,000 and it depreciates at a rate of 9% per year.

Answers

1. a. arithmetic

b. geometric, r=2

c. geometric, r=\frac{1}{7}

2. The first term is 3 and the common ratio is r=\frac{6}{3}=2 so a_n=3(2)^{n-1} .

The 20^{th} term is a_{20}=3(2)^{19}=1, 572, 864 .

3. The first term is -1024 and the common ratio is r=\frac{768}{-1024}=-\frac{3}{4} so a_n=-1024 \left(-\frac{3}{4}\right)^{n-1} .

Using the calculator sequence function to find the terms and MATH > Frac ,

seq \ (-1024(-3/4)^\land(x-1), x, 5, 11)= \left\{-324 \quad 243 \quad -\frac{729}{4} \quad \frac{2187}{16} \quad -\frac{6561}{256} \quad \frac{19683}{256} \quad -\frac{59049}{1024}\right\}

4. The first term (value of the car after 0 years) is $22,000. The common ratio is 1-.09 or 0.91 . The value of the car after n years can be determined by a_n=22, 000(0.91)^n . For 10 years we get a_{10}=22, 000(0.91)^{10}=8567.154599 \approx \$8567 .

Vocabulary

Geometric Sequence
A sequence in which the ratio of any two consecutive terms is constant.
Common Ratio
The value of the constant ratio between any two consecutive terms in a geometric sequence. Also, the value by which you multiply a term in the sequence to get the next term.

Explore More

Identify which of the following sequences are arithmetic, geometric or neither.

  1. 2, 4, 6, 8, \ldots
  2. \frac{1}{2}, \frac{3}{2}, \frac{9}{2}, \frac{27}{2}, \ldots
  3. 1, 2, 4, 7, \ldots
  4. 24, -16, \frac{32}{3}, -\frac{64}{9}, \ldots
  5. 10, 5, 0, -5, \ldots
  6. 3, 4, 7, 11, \ldots

Given the first term and common ratio, write the n^{th} term rule and use the calculator to generate the first five terms in each sequence.

  1. a_1=32 and r=\frac{3}{2}
  2. a_1=-81 and r=-\frac{1}{3}
  3. a_1=7 and r=2
  4. a_1=\frac{8}{125} and r=-\frac{5}{2}

Find the n^{th} term rule for each of the following geometric sequences.

  1. 162, 108, 72, \ldots
  2. -625, -375, -225, \ldots
  3. \frac{9}{4}, -\frac{3}{2}, 1, \ldots
  4. 3, 15, 75, \ldots
  5. 5, 10, 20, \ldots
  6. \frac{1}{2}, -2, 8, \ldots

Use a geometric sequence to solve the following word problems.

  1. Rebecca inherited some land worth $50,000 that has increased in value by an average of 5% per year for the last 5 years. If this rate of appreciation continues, about how much will the land be worth in another 10 years?
  2. A farmer buys a new tractor for $75,000. If the tractor depreciates in value by about 6% per year, how much will it be worth after 15 years?

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