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This page is devoted to discussing sequences using function notation.
Certain sequences (not all) can be defined (expressed) as an "explicit" formula.
An explicit formula will create a sequence using n, the number location of each term.
If you can find an explicit formula for a sequence, you will be able to quickly and easily find any term in the sequence simply by replacing n with the number of the term you seek.
An explicit formula designates the nth term of the sequence,
as an expression of n
(where n = the term's location). It defines the sequence as a formula in terms of n. It may be written in either subscript notation an, or in functional notation, f (n).
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 Sequence:
{10, 15, 20, 25, 30, 35, ...}. Find an explicit formula.
This example is an arithmetic sequence(the same number, 5, is added to each term to get to the next term).
Term Number |
Term |
Function Notation |
1 |
10
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f (1) |
2 |
15
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f (2) |
3 |
20
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f (3) |
4 |
25
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f (4) |
5 |
30
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f (5) |
6 |
35
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f (6) |
n |
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f (n) |
Explicit Formula:
in function notation: f (n) = 5n + 5 |
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This sequence is graphed in the first quadrant. Remember that the domain consists of the natural numbers, {1, 2, 3, ...}, and the range consists of the terms of the sequence. Notice that this sequence has a linear appearance. The rate of change between each of the points is "5 over 1". While the n value increases by a constant value of one, the f (n) value increases by a constant value of 5 (for this graph).
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It is easy to see that the explicit formula works once you are given the formula. Unfortunately, it is not always easy to come up with explicit formulas, when all you have is a list of the terms.
If your sequence is arithmetic, it will help if you look at the pattern of what is happening in the sequence. |
Explicit formula: f (n) = 10 + 5(n - 1) |
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If you compare the term number with how many times the common difference, 5, is added, you will see a pattern for an explicit formula:
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Now that you have the explicit formula, find the 100th term of this sequence.
Replace n with 100 in the explicit formula: f (n) = 10 + 5(n - 1)
f (100) = 10 + 5(100 - 1) = 10 + 5(99) = 10 + 495 = 505 ANSWER
 Sequence:
{3, 6, 12, 24, 48, 96, ...}. Find an explicit formula.
This example is a geometric sequence (the same number, 2, is multiplied times each term to get to the next term).
Term Number |
Term |
Function Notation |
1 |
3 |
f (1) |
2 |
6 |
f (2) |
3 |
12 |
f (3) |
4 |
24 |
f (4) |
5 |
48 |
f (5) |
6 |
96 |
f (6) |
n |
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f (n) |
Explicit Formula:
in function notation: f (n) = 3•(2)n-1
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Notice that this sequence has an exponential appearance. It may be the case with geometric sequences that the graph will increase (or decrease). The rate of change will increase (or decrease) as the value of n increases (it is not constant).
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Again, it is easy to see that a given explicit formula works. The problem is coming up with a formula when all you are given is a list of the terms.
If your sequence is geometric, it will help if you look at the pattern of what is happening in the sequence, in a manner similar to what we examined in the arithmetic sequence. |
Explicit formula: f (n) = 3 • 2n-1 |
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If you compare the term number with the powers of the common difference, 2, you will see a pattern for an explicit formula:
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Now that you have the explicit formula, find the 9th term of this sequence.
Replace n with 9 in the explicit formula: f (n) = 3 • 2n-1
f (9) = 3 • 29-1 = 3 • 28 = 3 • 256 = 768 ANSWER
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