# 11.1 Sequences and their notations  (Page 3/15)

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## Writing the terms of an alternating sequence defined by an explicit formula

Write the first five terms of the sequence.

${a}_{n}=\frac{{\left(-1\right)}^{n}{n}^{2}}{n+1}$

Substitute $n=1,$ $n=2,$ and so on in the formula.

$\begin{array}{lll}n=1\hfill & \begin{array}{cc}& \end{array}\hfill & {a}_{1}=\frac{{\left(-1\right)}^{1}{2}^{2}}{1+1}=-\frac{1}{2}\hfill \\ n=2\hfill & \begin{array}{cc}& \end{array}\hfill & {a}_{2}=\frac{{\left(-1\right)}^{2}{2}^{2}}{2+1}=\frac{4}{3}\hfill \\ n=3\hfill & \begin{array}{cc}& \end{array}\hfill & {a}_{3}=\frac{{\left(-1\right)}^{3}{3}^{2}}{3+1}=-\frac{9}{4}\hfill \\ n=4\hfill & \begin{array}{cc}& \end{array}\hfill & {a}_{4}=\frac{{\left(-1\right)}^{4}{4}^{2}}{4+1}=\frac{16}{5}\hfill \\ n=5\hfill & \hfill & {a}_{5}=\frac{{\left(-1\right)}^{5}{5}^{2}}{5+1}=-\frac{25}{6}\hfill \end{array}$

The first five terms are $\left\{-\frac{1}{2},\frac{4}{3},-\frac{9}{4},\frac{16}{5},-\frac{25}{6}\right\}.$

In [link] , does the (–1) to the power of $n$ account for the oscillations of signs?

Yes, the power might be $n,n+1,n-1,\text{\hspace{0.17em}}$ and so on, but any odd powers will result in a negative term, and any even power will result in a positive term.

Write the first five terms of the sequence:

${a}_{n}=\frac{4n}{{\left(-2\right)}^{n}}$

The first five terms are

## Investigating piecewise explicit formulas

We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions . Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.

Given an explicit formula for a piecewise function, write the first $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms of a sequence

1. Identify the formula to which $n=1$ applies.
2. To find the first term, $\text{\hspace{0.17em}}{a}_{1},\text{\hspace{0.17em}}$ use $\text{\hspace{0.17em}}n=1\text{\hspace{0.17em}}$ in the appropriate formula.
3. Identify the formula to which $\text{\hspace{0.17em}}n=2\text{\hspace{0.17em}}$ applies.
4. To find the second term, $\text{\hspace{0.17em}}{a}_{2},\text{\hspace{0.17em}}$ use $\text{\hspace{0.17em}}n=2\text{\hspace{0.17em}}$ in the appropriate formula.
5. Continue in the same manner until you have identified all $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ terms.

## Writing the terms of a sequence defined by a piecewise explicit formula

Write the first six terms of the sequence.

Substitute $\text{\hspace{0.17em}}n=1,n=2,\text{\hspace{0.17em}}$ and so on in the appropriate formula. Use ${n}^{2}$ when $n$ is not a multiple of 3. Use $\frac{n}{3}$ when $n$ is a multiple of 3.

The first six terms are

Write the first six terms of the sequence.

The first six terms are

## Finding an explicit formula

Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the $\text{\hspace{0.17em}}n\text{th}\text{\hspace{0.17em}}$ term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.

Given the first few terms of a sequence, find an explicit formula for the sequence.

1. Look for a pattern among the terms.
2. If the terms are fractions, look for a separate pattern among the numerators and denominators.
3. Look for a pattern among the signs of the terms.
4. Write a formula for ${a}_{n}$ in terms of $n.$ Test your formula for and $n=3.$

## Writing an explicit formula for the n Th term of a sequence

Write an explicit formula for the $n\text{th}$ term of each sequence.

1. $\left\{-\frac{2}{11},\frac{3}{13},-\frac{4}{15},\frac{5}{17},-\frac{6}{19},\dots \right\}$
2. $\left\{-\frac{2}{25}\text{,}-\frac{2}{125}\text{,}-\frac{2}{625}\text{,}-\frac{2}{3\text{,}125}\text{,}-\frac{2}{15\text{,}625}\text{,}\dots \right\}$
3. $\left\{{e}^{4}\text{,}{e}^{5}\text{,}{e}^{6}\text{,}{e}^{7}\text{,}{e}^{8}\text{,}\dots \right\}$

Look for the pattern in each sequence.

1. The terms alternate between positive and negative. We can use $\text{\hspace{0.17em}}{\left(-1\right)}^{n}\text{\hspace{0.17em}}$ to make the terms alternate. The numerator can be represented by $n+1.$ The denominator can be represented by $2n+9.$

${a}_{n}=\frac{{\left(-1\right)}^{n}\left(n+1\right)}{2n+9}$

2. The terms are all negative.

So we know that the fraction is negative, the numerator is 2, and the denominator can be represented by ${5}^{n+1}.$

${a}_{n}=-\frac{2}{{5}^{n+1}}$
3. The terms are powers of $e.$ For $n=1,$ the first term is ${e}^{4}$ so the exponent must be $n+3.$

${a}_{n}={e}^{n+3}$

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris