11.1 Sequences and their notations (Page 2/15)

Precalculus

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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, \dots, 2^{n}, \dots} .

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, \dots, 2^{n}, \dots, 1024} .

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

A General Note

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}, \dots, a_{n}, \dots

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 .

Q&A

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.

How To

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

Substitute each value of $n$ into the formula. Begin with $n = 1$ to find the first term, $a_{1} .$
To find the second term, $a_{2},$ use $n = 2.$
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 $n .$

\begin{array}{l} 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 \end{array}

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

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Try It

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.

How To

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

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.
To find the second term, $a_{2},$ use $n = 2.$
Continue in the same manner until you have identified all $n$ terms.

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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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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