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n 1 2 3 4 5 n
n th term of the sequence, a n 2 4 8 16 32 2 n

Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in [link] that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.

Graph of a plotted exponential function, f(n) = 2^n, where the x-axis is labeled n and the y-axis is labeled a_n.

Lastly, we can write this particular sequence as

{ 2 , 4 , 8 , 16 , 32 , , 2 n , } .

A sequence that continues indefinitely is called an infinite sequence . The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write

{ 2 , 4 , 8 , 16 , 32 , , 2 n , , 1024 } .

This sequence is called a finite sequence because it does not continue indefinitely.

Sequence

A sequence    is a function whose domain is the set of positive integers. A finite sequence    is a sequence whose domain consists of only the first n positive integers. The numbers in a sequence are called terms . The variable a with a number subscript is used to represent the terms in a sequence and to indicate the position of the term in the sequence.

a 1 , a 2 , a 3 , , a n ,

We call a 1 the first term of the sequence, a 2 the second term of the sequence, a 3 the third term of the sequence, and so on. The term a n is called the n th term of the sequence , or the general term of the sequence. An explicit formula    defines the n th term of a sequence using the position of the term. A sequence that continues indefinitely is an infinite sequence    .

Does a sequence always have to begin with a 1 ?

No. In certain problems, it may be useful to define the initial term as a 0 instead of a 1 . In these problems, the domain of the function includes 0.

Given an explicit formula, write the first n terms of a sequence.

  1. Substitute each value of n into the formula. Begin with n = 1 to find the first term, a 1 .
  2. To find the second term, a 2 , use n = 2.
  3. Continue in the same manner until you have identified all n terms.

Writing the terms of a sequence defined by an explicit formula

Write the first five terms of the sequence defined by the explicit formula a n = 3 n + 8.

Substitute n = 1 into the formula. Repeat with values 2 through 5 for n .

n = 1 a 1 = 3 ( 1 ) + 8 = 5 n = 2 a 2 = 3 ( 2 ) + 8 = 2 n = 3 a 3 = 3 ( 3 ) + 8 = 1 n = 4 a 4 = 3 ( 4 ) + 8 = 4 n = 5 a 5 = 3 ( 5 ) + 8 = 7

The first five terms are { 5 , 2 , −1 , −4 , −7 } .

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Write the first five terms of the sequence defined by the explicit formula     t n = 5 n 4.

The first five terms are { 1 , 6 ,   11 ,   16 ,   21 } .

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Investigating alternating sequences

Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as n increases. Let’s take a look at the following sequence.

{ 2 , −4 , 6 , −8 }

Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.

Given an explicit formula with alternating terms, write the first n terms of a sequence.

  1. Substitute each value of n into the formula. Begin with n = 1 to find the first term, a 1 . The sign of the term is given by the ( 1 ) n in the explicit formula.
  2. To find the second term, a 2 , use n = 2.
  3. Continue in the same manner until you have identified all n terms.

Questions & Answers

find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
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Opoku
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Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
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Vedant
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Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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salma
Practice Key Terms 8

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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