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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

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 8

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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