11.1 Sequences and their notations (Page 3/15)

Precalculus

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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{{(- 1)}^{n} n^{2}}{n + 1}

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

\begin{array}{l} n = 1 & \begin{matrix}  \end{matrix} & a_{1} = \frac{{(- 1)}^{1} 2^{2}}{1 + 1} = - \frac{1}{2} \\ n = 2 & \begin{matrix}  \end{matrix} & a_{2} = \frac{{(- 1)}^{2} 2^{2}}{2 + 1} = \frac{4}{3} \\ n = 3 & \begin{matrix}  \end{matrix} & a_{3} = \frac{{(- 1)}^{3} 3^{2}}{3 + 1} = - \frac{9}{4} \\ n = 4 & \begin{matrix}  \end{matrix} & a_{4} = \frac{{(- 1)}^{4} 4^{2}}{4 + 1} = \frac{16}{5} \\ n = 5 & a_{5} = \frac{{(- 1)}^{5} 5^{2}}{5 + 1} = - \frac{25}{6} \end{array}

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

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Q&A

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

Yes, the power might be $n, n + 1, n - 1,$ and so on, but any odd powers will result in a negative term, and any even power will result in a positive term.

Try It

Write the first five terms of the sequence:

a_{n} = \frac{4 n}{{(- 2)}^{n}}

The first five terms are ${- 2, 2, - \frac{3}{2}, 1, - \frac{5}{8}} .$

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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.

How To

Given an explicit formula for a piecewise function, write the first $n$ terms of a sequence

Identify the formula to which $n = 1$ applies.
To find the first term, $a_{1},$ use $n = 1$ in the appropriate formula.
Identify the formula to which $n = 2$ applies.
To find the second term, $a_{2},$ use $n = 2$ in the appropriate formula.
Continue in the same manner until you have identified all $n$ terms.

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

Write the first six terms of the sequence.

a_{n} = {\begin{array}{l} n^{2} & if n is not divisible by 3 \\ \frac{n}{3} & if n is divisible by 3 \end{array}

Substitute $n = 1, n = 2,$ 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.

\begin{array}{l} a_{1} = 1^{2} = 1 \begin{matrix}  \end{matrix} & 1 is not a multiple of 3 .  Use n^{2} . \\ a_{2} = 2^{2} = 4 & 2 is not a multiple of 3 .  Use n^{2} . \\ a_{3} = \frac{3}{3} = 1 & 3 is a multiple of 3 .  Use \frac{n}{3} . \\ a_{4} = 4^{2} = 16 & 4 is not a multiple of 3 .  Use n^{2} . \\ a_{5} = 5^{2} = 25 & 5 is not a multiple of 3 .  Use n^{2} . \\ a_{6} = \frac{6}{3} = 2 & 6 is a multiple of 3 .  Use \frac{n}{3} . \end{array}

The first six terms are ${1, 4, 1, 16, 25, 2} .$

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Try It

Write the first six terms of the sequence.

a_{n} = {\begin{array}{l} 2 n^{3} & if n is odd \\ \frac{5 n}{2} & if n is even \end{array}

The first six terms are ${2, 5, 54, 10, 250, 15} .$

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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 $n th$ 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.

How To

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

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

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

Write an explicit formula for the $n th$ term of each sequence.

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

Look for the pattern in each sequence.

The terms alternate between positive and negative. We can use ${(- 1)}^{n}$ to make the terms alternate. The numerator can be represented by $n + 1.$ The denominator can be represented by $2 n + 9.$
$a_{n} = \frac{{(- 1)}^{n} (n + 1)}{2 n + 9}$
The terms are all negative.

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

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

$a_{n} = e^{n + 3}$

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?

ADNAN Reply

linear speed of an object

Melissa Reply

an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object

Melissa

test

Matrix

how to find domain

Mohamed Reply

like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.

Dan

define the term of domain

Moha

if a>0 then the graph is concave

Angel Reply

if a<0 then the graph is concave blank

Angel

what's a domain

Kamogelo Reply

The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.

Spiro

how fast can i understand functions without much difficulty

Joe Reply

what is inequalities

Nathaniel

functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2

Dan

what is set?

Kelvin Reply

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size

Divya Reply

I got 300 minutes. is it right?

Patience

no. should be about 150 minutes.

Jason

It should be 158.5 minutes.

ok, thanks

Patience

100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes

Thomas

158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49

Dan

what is the importance knowing the graph of circular functions?

Arabella Reply

can get some help basic precalculus

ismail Reply

What do you need help with?

Andrew

how to convert general to standard form with not perfect trinomial

Camalia Reply

can get some help inverse function

ismail

Rectangle coordinate

Asma Reply

how to find for x

Jhon Reply

it depends on the equation

Robert

yeah, it does. why do we attempt to gain all of them one side or the other?

Melissa

how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2

Dan

whats a domain

mike Reply

The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.

Spiro

Spiro; thanks for putting it out there like that, 😁

Melissa

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.

Churlene Reply

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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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