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9.1 Sequences and their notations  (Page 2/15)

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n 1 2 3 4 5 n
n 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.

Graph of a plotted exponential function, f(n) = 2^n, where the x-axis is labeled n and the y-axis is labeled a_n.

Lastly, we can write this particular sequence as

  { 2 , 4 , 8 , 16 , 32 , … , 2 n , … } .

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

  { 2 , 4 , 8 , 16 , 32 , … , 2 n , … , 1024 } .

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 , … , a n , …

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 th term of the sequence , or the general term of the sequence. An explicit formula    defines the n 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   a 1 ?

No. In certain problems, it may be useful to define the initial term as a 0 instead of   a 1 .   In these problems, the domain of the function includes 0.

Given an explicit formula, 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 .
  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 = − 3 n + 8.

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

n = 1 a 1 = − 3 ( 1 ) + 8 = 5 n = 2 a 2 = − 3 ( 2 ) + 8 = 2 n = 3 a 3 = − 3 ( 3 ) + 8 = − 1 n = 4 a 4 = − 3 ( 4 ) + 8 = − 4 n = 5 a 5 = − 3 ( 5 ) + 8 = − 7

The first five terms are   { 5 ,   2 ,   −1 ,   −4 ,   −7 } .

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Write the first five terms of the sequence defined by the explicit formula       t n = 5 n − 4.

The first five terms are { 1 , 6 ,   11 ,   16 ,   21 } .

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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.

{ 2 , −4 , 6 , −8 }

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 ( − 1 ) n in the explicit formula.
  2. To find the second term,   a 2 ,   use   n = 2.  
  3. Continue in the same manner until you have identified all n terms.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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