# 9.1 Sequences and their notations  (Page 7/15)

 Page 7 / 15

$\left(\frac{12}{6}\right)!$

$\frac{12!}{6!}$

$665,280$

$\frac{100!}{99!}$

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

${a}_{n}=\frac{n!}{{n}^{\text{2}}}$

First four terms: $1,\frac{1}{2},\frac{2}{3},\frac{3}{2}$

${a}_{n}=\frac{3\cdot n!}{4\cdot n!}$

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

First four terms: $-1,2,\frac{6}{5},\frac{24}{11}$

${a}_{n}=\frac{100\cdot n}{n\left(n-1\right)!}$

## Graphical

For the following exercises, graph the first five terms of the indicated sequence

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

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

For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

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

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

## Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

• On the home screen, key in the value for the initial term $\text{\hspace{0.17em}}{a}_{1}\text{\hspace{0.17em}}$ and press [ENTER] .
• Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term $\text{\hspace{0.17em}}{a}_{n-1}.\text{\hspace{0.17em}}$ Press [ENTER] .
• Continue pressing [ENTER] to calculate the values for each successive term.

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence Use the> Frac feature to give fractional results.

First five terms: $\frac{29}{37},\frac{152}{111},\frac{716}{333},\frac{3188}{999},\frac{13724}{2997}$

Find the 15 th term of the sequence

Find the first five terms of the sequence

First five terms: $2,3,5,17,65537$

Find the first ten terms of the sequence

Find the tenth term of the sequence

${a}_{10}=7,257,600$

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

• In the home screen, press [2ND] LIST .
• Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
• In the line headed “Expr:” type in the explicit formula, using the $\text{\hspace{0.17em}}\left[\text{X,T},\theta ,n\right]\text{\hspace{0.17em}}$ button for $\text{\hspace{0.17em}}n$
• In the line headed “Variable:” type in the variable used on the previous step.
• In the line headed “start:” key in the value of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ that begins the sequence.
• In the line headed “end:” key in the value of $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ that ends the sequence.
• Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

Using a TI-83, do the following.

• In the home screen, press [2ND] LIST .
• Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
• Enter the items in the order “Expr” , “Variable” , “start” , “end” separated by commas. See the instructions above for the description of each item.
• Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first five terms of the sequence ${a}_{n}=-\frac{28}{9}n+\frac{5}{3}.$

List the first six terms of the sequence

First six terms: $0.042,0.146,0.875,2.385,4.708$

List the first five terms of the sequence ${a}_{n}=\frac{15n\cdot {\left(-2\right)}^{n-1}}{47}$

List the first four terms of the sequence ${a}_{n}={5.7}^{n}+0.275\left(n-1\right)!$

First four terms: $5.975,32.765,185.743,1057.25,6023.521$

List the first six terms of the sequence ${a}_{n}=\frac{n!}{n}.$

## Extensions

Consider the sequence defined by ${a}_{n}=-6-8n.$ Is ${a}_{n}=-421$ a term in the sequence? Verify the result.

If $\text{\hspace{0.17em}}{a}_{n}=-421\text{\hspace{0.17em}}$ is a term in the sequence, then solving the equation $-421=-6-8n$ for $n$ will yield a non-negative integer. However, if $\text{\hspace{0.17em}}-421=-6-8n,\text{\hspace{0.17em}}$ then $n=51.875$ so ${a}_{n}=-421$ is not a term in the sequence.

What term in the sequence ${a}_{n}=\frac{{n}^{2}+4n+4}{2\left(n+2\right)}$ has the value $41?$ Verify the result.

Find a recursive formula for the sequence ( Hint : find a pattern for $\text{\hspace{0.17em}}{a}_{n}\text{\hspace{0.17em}}$ based on the first two terms.)

${a}_{1}=1,{a}_{2}=0,{a}_{n}={a}_{n-1}-{a}_{n-2}$

Calculate the first eight terms of the sequences ${a}_{n}=\frac{\left(n+2\right)!}{\left(n-1\right)!}$ and ${b}_{n}={n}^{3}+3{n}^{2}+2n,$ and then make a conjecture about the relationship between these two sequences.

Prove the conjecture made in the preceding exercise.

$\frac{\left(n+2\right)!}{\left(n-1\right)!}=\frac{\left(n+2\right)·\left(n+1\right)·\left(n\right)·\left(n-1\right)·...·3·2·1}{\left(n-1\right)·...·3·2·1}=n\left(n+1\right)\left(n+2\right)={n}^{3}+3{n}^{2}+2n$

#### Questions & Answers

12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8