We saw in Basic Information, that some sequences can be expressed by a single formula.
Such sequences are referred to as explicit sequences.

Explicit Sequences:
Example:  an = 5n + 5

Certain sequences (not all) can be defined (expressed) as an "explicit" formula that defines the pattern of the sequence.

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 is most commonly written in subscript notation, an.

In plain English, an explicit formula establishes a formula for finding any term in a sequence,
by using its "position number" in the sequence.
An explicit formula allows you to jump to any term in a sequence to find its value.

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Let's take a look at some sequences that are explicit and some that are not.

Ex2

{1, 3, 5, 7, 9, 11, . . .}

This is the set of odd, positive integers.
Pattern: the number 2 is added to each term.
It can be expressed as
an = 2n - 1

Explicit Sequence

Find the 20th term.
a20 = 2(20) - 1
a20 = 39
Notice how the graph is linear (a straight line).

 

Ex2

{8, 6, 4, 2, 0, 2, 4, . . .}


Pattern: the positive values in this sequence decreased, then increased, by 2.

 It can be expressed as
an = 2 | n - 5 |

Explicit Sequence

Find the 100th term.
a100 = 2 | 100 - 5 |
a100 = 190

The pattern of decreasing then increasing positive values is created by the absolute value at work.
The absolute value graph shows how the values remain positive (or 0).


Ex2

{-9, -6, -3, 0, 3, 6, . . .}

Increasing sequence.

Pattern: the number 3 is added to each term.
It can be expressed as
an = 3n - 12

Explicit Sequence

Is 43 a term in this sequence?
 an = 3n - 12 = 43
3n = 55
n = 18.3333333
Nope! n must be a positive integer.
Notice how this sequence's graph goes into Quadrant IV.
The x-values remain positive integers, but the y-values include negative entries.


Ex2



Pattern: the numerators are a constant 1, and the denominators are increasing by an addition of 1 (starting with 2)..

It can be expressed as

Explicit Sequence

If an = 1/32, what is the value of n?
an = 1/(1+n) = 1/32
1 + n = 32
n = 31     (31st term)


Ex2


{3, 1, 4, 1, 5, 9, . . .}

The digits in π.

There is no formula for the
nth term of this sequence.

NOT Explicit

 

Ex2

{g, o, o, d, f, a, n}

The letters in "good fan".

There is no formula for the
nth term of this sequence.

NOT Explicit
The graph is drawn using the number of the letters placement in the alphabet ( a, b, c, ... is 1, 2, 3, ...)



Ex2


The sequence alternates between positive and negative values.

Pattern: the numerators are a constant 1, and the denominators are increasing by an addition of 2 (starting with 5). 		It can be expressed as

Explicit Sequence

Find the 12th term.
a12 = (-1)13/ (2(12) +3)
a12 = -1/27
Notice that the (-1)n+1 is creating the alternating of positive and negative terms.

 

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scratch head
 It is easy to see that an explicit formula works nicely
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.
The key to finding a formula is finding a pattern.

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ti84c
For calculator
help with
sequences
click here.

Arrow down to
"In Func MODE"

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