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13.1 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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