# 13.1 Sequences and their notations  (Page 2/15)

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 $n$ 1 2 3 4 5 $n$ $n\text{th}$ 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.

Lastly, we can write this particular sequence as

$\text{\hspace{0.17em}}\left\{2,4,8,16,32,\dots ,{2}^{n},\dots \right\}.$

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

$\text{\hspace{0.17em}}\left\{2,4,8,16,32,\dots ,{2}^{n},\dots ,1024\right\}.$

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.

${a}_{1},{a}_{2},{a}_{3},\dots ,{a}_{n},\dots$

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 $n\text{th}$ term of the sequence , or the general term of the sequence. An explicit formula    defines the $n\text{th}$ 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 $\text{\hspace{0.17em}}{a}_{1}?$

No. In certain problems, it may be useful to define the initial term as ${a}_{0}$ instead of $\text{\hspace{0.17em}}{a}_{1}.\text{\hspace{0.17em}}$ In these problems, the domain of the function includes 0.

Given an explicit formula, write the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ 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 ${a}_{n}=-3n+8.$

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

$\begin{array}{llllll}n=1\hfill & \hfill & \hfill & \hfill & \hfill & {a}_{1}=-3\left(1\right)+8=5\hfill \\ n=2\hfill & \hfill & \hfill & \hfill & \hfill & {a}_{2}=-3\left(2\right)+8=2\hfill \\ n=3\hfill & \hfill & \hfill & \hfill & \hfill & {a}_{3}=-3\left(3\right)+8=-1\hfill \\ n=4\hfill & \hfill & \hfill & \hfill & \hfill & {a}_{4}=-3\left(4\right)+8=-4\hfill \\ n=5\hfill & \hfill & \hfill & \hfill & \hfill & {a}_{5}=-3\left(5\right)+8=-7\hfill \end{array}$

The first five terms are $\text{\hspace{0.17em}}\left\{5,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}-1,\text{\hspace{0.17em}}-4,\text{\hspace{0.17em}}-7\right\}.$

Write the first five terms of the sequence defined by the explicit formula     $\text{\hspace{0.17em}}{t}_{n}=5n-4.$

The first five terms are

## 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.

$\left\{2,-4,6,-8\right\}$

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 ${\left(-1\right)}^{n}$ in the explicit formula.
2. To find the second term, $\text{\hspace{0.17em}}{a}_{2},\text{\hspace{0.17em}}$ use $\text{\hspace{0.17em}}n=2.\text{\hspace{0.17em}}$
3. Continue in the same manner until you have identified all $n$ terms.

#### Questions & Answers

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
Jeffrey