# 13.1 Sequences and their notations  (Page 6/15)

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## Verbal

Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?

A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.

Describe three ways that a sequence can be defined.

Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.

Yes, both sets go on indefinitely, so they are both infinite sequences.

What happens to the terms ${a}_{n}$ of a sequence when there is a negative factor in the formula that is raised to a power that includes $n?$ What is the term used to describe this phenomenon?

What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.

A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out $\text{13}\cdot \text{12}\cdot \text{11}\cdot \text{10}\cdot \text{9}\cdot \text{8}\cdot \text{7}\cdot \text{6}\cdot \text{5}\cdot \text{4}\cdot \text{3}\cdot \text{2}\cdot \text{1}\text{.}$

## Algebraic

For the following exercises, write the first four terms of the sequence.

${a}_{n}={2}^{n}-2$

${a}_{n}=-\frac{16}{n+1}$

First four terms:

${a}_{n}=-{\left(-5\right)}^{n-1}$

${a}_{n}=\frac{{2}^{n}}{{n}^{3}}$

First four terms: .

${a}_{n}=\frac{2n+1}{{n}^{3}}$

${a}_{n}=1.25\cdot {\left(-4\right)}^{n-1}$

First four terms: .

${a}_{n}=-4\cdot {\left(-6\right)}^{n-1}$

${a}_{n}=\frac{{n}^{2}}{2n+1}$

First four terms: .

${a}_{n}={\left(-10\right)}^{n}+1$

${a}_{n}=-\left(\frac{4\cdot {\left(-5\right)}^{n-1}}{5}\right)$

First four terms:

For the following exercises, write the first eight terms of the piecewise sequence.

$-0.6,-3,-15,-20,-375,-80,-9375,-320$

For the following exercises, write an explicit formula for each sequence.

${a}_{n}={n}^{2}+3$

$-4,2,-10,14,-34,\dots$

$1,1,\frac{4}{3},2,\frac{16}{5},\dots$

$0,\frac{1-{e}^{1}}{1+{e}^{2}},\frac{1-{e}^{2}}{1+{e}^{3}},\frac{1-{e}^{3}}{1+{e}^{4}},\frac{1-{e}^{4}}{1+{e}^{5}},\dots$

$1,-\frac{1}{2},\frac{1}{4},-\frac{1}{8},\frac{1}{16},\dots$

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

For the following exercises, write the first five terms of the sequence.

First five terms:

First five terms:

For the following exercises, write the first eight terms of the sequence.

For the following exercises, write a recursive formula for each sequence.

$-2.5,-5,-10,-20,-40,\dots$

$-8,-6,-3,1,6,\dots$

${a}_{1}=-8,{a}_{n}={a}_{n-1}+n$

${a}_{1}=35,{a}_{n}={a}_{n-1}+3$

$15,3,\frac{3}{5},\frac{3}{25},\frac{3}{125},\cdots$

For the following exercises, evaluate the factorial.

$6!$

$720$

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
circular region of radious
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions