**Applications of Geometric Sequences and Series**

A lot of problems can be solved by the formulas for the general term of a geometric sequence and geometric series, finite or infinite. Of these applications, that of the infinite geometric series is most interesting as seen in the examples that follow.

**Chain Letter Problem
Sierpinski’s Triangle
Binary System and an an Amount of Memory
Changing Repeating Decimals to Fractions
Growth of Bacteria
Rebounding Balls
Tax Rebates and the Multiplier Effect, Computing a Lifetime Salary, Allowance**

This section consists of a mixture of problems where the work covered in the first five exercises is applied to a variety of situations.

The following general guidelines can assist you in solving the problems.

1. Read the question carefully.

2. Decide whether the information suggests an arithmetic or geometric sequence. Check to see if there is a constant difference between successive terms or a constant ratio. If there is neither, look for a simple number pattern such as the difference between successive terms changing in a regular way.

3. Write the information from the problem using appropriate notation. For example, if you are told that the fifth term is 12, write *t*_{5}=12. If the sequence is arithmetic, you then have an equation to work with, namely: a+4d=12. If you know the sequence is geometric, then *a⋅r*^{4}=12.

4. Define what you have to calculate and write an appropriate formula or formulas. For example, if you have to find the tenth number in a sequence that you know is geometric, you have an equation: *t*_{10}=*a⋅r*^{9}. This can be calculated if a and r are known or can be established.

5. Use algebra to find what is required in the problem.

📌 (Chain E-Mail). Melina receives a chain e-mail that she forwards to 7 of her friends. Each of her friends forwards it to 7 of their friends.

a. Write an explicit formula for the pattern.

b. How many will receive the e-mail after 6 forwards?

✍ Solution:

a. Melina receives a chain e-mail, forwards it to 7 friends, and each friend forwards it to 7 friends.

Therefore,

*a*

_{1}=1,

*a*

_{2}=7, and

*a*

_{3}=49. The common ratio is 7.

For an explicit formula, substitute

*a*

_{1}=1 and r=7 in the nth term formula.

*a*=

_{n}*a*

_{1}

*r*

^{n-1}

=1⋅7

^{n-1}or 7

^{n-1}

b. Use the explicit formula you found in part (a.) to find

*a*

_{6}.

*a*=7

_{n}^{n-1}

*a*

_{6}=7

^{6-1}or 7

^{5}

=16,807

Therefore, after 6 forwards 16,807 people will have received the e-mail.

📌 (Sierpinski’s Triangle). Consider the inscribed equilateral triangles shown. The perimeter of each triangle is one half of the perimeter of the next larger triangle. What is the perimeter of the smallest triangle?

✍ Solution:

This is a geometric sequence. The first term is 3(40) or 120 and the common ratio is ½. To find the perimeter of the smallest triangle, find the 5th term of the sequence.

*a*=

_{n}*a*

_{1}

*r*

^{n-1}

*a*

_{5}=120⋅½

^{5-1}

=120⋅½

^{4}

=7.5

The perimeter of the smallest triangle is 7.5 centimeters.

**💪 Binary System and an Amount of Memory
💔 Changing Repeating Decimals to Fractions
🐞 Growth of Bacteria
🏀 Rebounding Balls
Tax Rebates and the Multiplier Effect, Computing a Lifetime Salary, Allowance 👀**