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A sequence is a set of numbers in numerical order. You may be familiar with arithmetic sequences, which have a common difference between each term, a constant number that is either added or subtracted as we go from one term to the next.
EXAMPLE
Consider the arithmetic sequence , with a common ratio of 2, where we add 2 to get the value of the next term: 3 plus 2 is 5, 5 plus 2 is 7, 7 plus 2 is 9, etc.With geometric sequences, there is a common ratio between each term. Similar to arithmetic sequences, the common ratio is a constant value; however it is not added or subtracted from one term to the next, it is multiplied by the preceding term to continue the sequence.
EXAMPLE
Consider the geometric sequence , with a common ratio of 2. We can multiply a term by 2 to get the value of the next term: 3 times 2 is 6, 6 times 2 is 12, 12 times 2 is 48, etc. The next term in the sequence would be 96, and 192 would follow that.The previous sequence was also an example of a divergent sequence. A divergent sequence is a sequence whose terms do not have a finite limit; they tend toward positive or negative infinity.
Since we are continuously multiplying each term in the sequence by 2 in the above example, we know that the terms will continue to increase in value, making the limit of values positive infinity. Even if the common ratio were -2, the terms would alternate between positive and negative, but would still grow in absolute magnitude. Divergent sequences are characterized by a common ratio greater than 1 or less than -1.
EXAMPLE
Consider if we have a geometric sequence with a common ratio of 5. The terms will tend toward positive infinity as we continue:In contrast, some geometric sequences are convergent. The terms in convergent sequences do have a limit. Convergent sequences are characterized by a common ratio between -1 and 1.
EXAMPLE
Consider if we have a geometric sequence with a common ratio of 0.5, as in the example below:Let's return to the geometric sequence: . It was easy enough to find the value of the next two terms, by multiplying 48 by 2 to get 96, and then by 2 again to get 192. What if we wanted to find the value of the 78th term? Or even the 115th therm? Certainly finding the value of that term by continuously multiplying by 2 is inefficient.
To find the nth term of a sequence, we can use the geometric sequence formula:
In this formula:
EXAMPLE
Consider the geometric sequence .EXAMPLE
Use the above formula to find the value of the 7th term in the sequence.Plug in 7 for n | |
Evaluate the exponent | |
Raise to the 6th power | |
Evaluate fraction | |
Multiply 8748 and | |
Divide 8749 by 729 | |
Our solution |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License