Geometric Sequences and Series | ||||||||||||||||||||||||||||

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Introduction | ||||||||||||||||||||||||||||

This page will teach you about geometric sequences and series. These are the sections within this page:
- Identifying Geometric Sequences
- Formulas for the Nth Term: Recursive and Explicit Rules
- Calculating the nth Term of Geometric Sequences
- Finding the Number of Terms in a Geometric Sequence
- Finding the Sum of Geometric Series
- Instructional videos
- Interactive Quizmasters
- Related Lessons and Quizzes
- Assignment Expert – Calculus Homework Help
MATHguide has a video with guided notes to help you understand geometric sequences. Access the guided notes and then proceed with the video. This activity/video addresses two Common Core State Standards: HSF.BF.A.1.A and HSF.BF.A.2. ideo: Geometric Sequences ctivity: Geometric Sequences: Guided Notes
If you rather, you can continue on with this text-based lesson to learn about geometric sequences and series.
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Sequences of numbers that follow a pattern of multiplying a fixed number from one term to the next are called geometric sequences. The following sequences are geometric sequences: Sequence A: 1 , 2 , 4 , 8 , 16 , ... For sequence A, if we multiply by 2 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number times 2 is the third number: 2 × 2 = 4, and so on.
For sequence B, if we multiply by 6 to the first number we will get the second number. This also works for any pair of consecutive numbers. The third number times 6 is the fourth number: 0.36 × 6 = 2.16, which will work throughout the entire sequence. Sequence C is a little different because it seems that we are dividing; yet to stay consistent with the theme of geometric sequences, we must think in terms of multiplication. We need to multiply by -1/2 to the first number to get the second number. This too works for any pair of consecutive numbers. The fourth number times -1/2 is the fifth number: -2 × -1/2 = 1.
Because these sequences behave according to this simple rule of multiplying a constant number to one term to get to another, they are called geometric sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called
Mathematicians also refer to generic sequences using the letter
This means that if we refer to the tenth term of a certain sequence, we will label it a
The
...where n is any positive integer greater than 1.
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To determine any number within a geometric sequence, there are two formulas that can be utilized. Here is the recursive rule.
The recursive rule means to find any number in the sequence, we must multiply the Let us say we were given this geometric sequence.
We can see that the
What if we wanted to find the 20th number? We would have to find all off the numbers before the 20th number, if we use the
To find the 20th number, all we have to do is multiply the first number in the sequence by the common ratio raised to the 19th power, like this. _{n} = (a_{1})r^{n-1}a _{20} = (1)(3)^{20-1}a _{20} = (3)^{19}a _{20} = 11622615explicit rule is far superior to the recursive rule.
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In this section, we will look at coming up with a unique formula to define all the terms in a geometric sequence. We will use the explicit rule to help us.
We will look at two examples to explain this skill.
The fixed number, called the r-value or _{n} = (a_{1})r^{n-1}a _{n} = (1)(2)^{n-1}a _{n} = 2^{n-1}
_{n} = 2^{n-1}a _{9} = 2^{9-1}a _{9} = 2^{8}a _{9} = 256
_{n} = 2^{n-1} and the 9th term is 256.
The _{n} = (a_{1})r^{n-1}a _{n} = (0.01)(6)^{n-1}
_{n} = (0.01)(6)^{n-1}a _{15} = (0.01)(6)^{15-1}a _{15} = (0.01)(6)^{14}a _{15} = 78364164_{n} = (0.01)(6)^{n-1} and the 15th term is 78,364,164.
uizmaster: Finding Formula for General Term
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Let us look at an example where we are given a sequence of numbers and we have to find how many numbers there are in the sequences.
First, we will develop a formula for this sequence using the _{n} = (a_{1})r^{n-1}a _{n} = (1)(2)^{n-1}a _{n} = 2^{n-1}
_{n}. Using the formula, we get this.
_{n} = 2^{n-1}256 = 2 ^{n-1}
^{n-1}log(256) = log(2 ^{n-1})log(256) = (n - 1)log(2) log(256)/log(2) = (n - 1) log(256)/log(2) + 1 = n 9 = n n = 9 | ||||||||||||||||||||||||||||

Given our generic geometric sequence...
...we can look at it as a series.
As we can see, the only difference between a sequence and a series is that a sequence is a list of numbers and a series is a sum of numbers. There exists a formula that can add a finite list of numbers and a formula for an infinite list of numbers. Here are the formulas... _{n} is the sum of the first n numbers, a_{1} is the first number in the sequence, r is the common ratio of the sequence, and -1 < r < 1 for infinite series. Let's use examples to investigate both formulas.
The sum formula requires us to know the first term [a
We can see a
ideo: Sum of a Finite Geometric Series
The only way we can add an infinite series is for two conditions to be met: a) it has to be a geometric series and b) the
Looking at the series, we can see that there is a ^{2}/_{3}.
We need to check the conditions to see if we can use the infinite sum formula. It does have a
ideo: Sum of an Infinite Geometric Series |

Here are the available videos for this content:
ideo: Geometric Sequences |

After reading the lesson, try our quizmaster. MATHguide has developed numerous testing and checking programs to solidify skills demonstrated in this lesson. The following quizmasters are available:
uizmaster: Finding Formula for General Term |

After reading the lesson, try a related lesson.
esson: Arithmetic Sequences and Series
uiz: Determine the Function |