(How). Are arithmetic sequences related to roofing?
A roofer is nailing shingles to the roof of a house in overlapping rows. There are three shingles in the top row. Since the roof widens from top to bottom, one additional shingle is needed in each successive row.
ARITHMETIC SEQUENCES
The numbers 3, 4, 5, 6, …, representing the number of shingles in each row, are an example of a sequence of numbers. A sequence is a list of numbers in a particular order. Each number in a sequence is called a term. The first term is symbolized by a1, the second term is symbolized by a2, and so on.
The graph represents the information from the table above. A sequence is a discrete function whose domain is the set of positive integers.
Many sequences have patterns. For example, in sequence above for the number of shingles, each term can be found by adding 1 to the previous term. A sequence of this type is called an arithmetic sequence.
An arithmetic sequence is a sequence in which each term after the first is found by adding a constant, called the common differenced, to the previous term.
The set of positive odd numbers, 1, 3, 5, 7, … , is a sequence. The first term, a1, is 1 and each term is 2 greater than the preceding term. The difference between consecutive terms is 2. We say that 2 is the common difference for the sequence. The set of positive odd numbers is an example an arithmetic sequence.
DEFINITION An arithmetic sequence is a sequence such that for all n, there is a constant d such that an–a(n-1) =d.
For an arithmetic sequence, the recursive formula is:
an=a(n-1) +d
An arithmetic sequence is formed when each term after the first is obtained by adding the same constant to the previous term. For example, look at the first five terms of the sequence of positive odd numbers.
a1=1 a1=1
a2=a1+2=1+2=3 a2=1+1(2)
a3=a2+2=3+2=5 a3=[1+1(2)]+2=1+2(2)
a4=a3+2=5+2=7 a4=[1+2(2)]+2=1+3(2)
a5=a4+2=7+2=9 a5=[1+3(2)]+2=1+4(2)
For this sequence, a1=1 and each term after the first is found by adding 2 to the preceding term. Therefore, for each term, 2 has been added to the first term one less time than the number of the term. For the second term, 2 has been added once; for the third term, 2 has been added twice; for the fourth term, 2 has been added three times. In general, for the nth term, 2 has been added n-1 times.
Shown below is a general arithmetic sequence with a1 as the first term and d as the common difference:
If the first term of an arithmetic sequence is al and the common difference is d, then for each term of the sequence, d has been added to all one less time than the number of the term.
Therefore, any term an of an arithmetic sequence can be evaluated with the formula
an=a1+(n-1)d
where d is the common difference of the sequence.
Arithmetic sequences
At a racetrack a new prototype racing car unfortunately develops an oil leak. Each second, a drop of oil hits the road. The driver of the car puts her foot on the accelerator and the car increases speed at a steady rate as it hurtles down the straight. The diagram below shows the pattern of oil drops on the road with the distances between the drops labelled.
The sequence of distances travelled in metres each second is {10, 18, 26, 34, 42, …}. The first term in the sequence, t1, is 10, and as you can see, each subsequent term is 8 more than the previous term. This type of sequence is given a special name – an arithmetic sequence.
An arithmetic sequence is a sequence where there is a common difference between any two successive terms.
We can list the sequence in a table as shown in table A below. From this table we can see that it is possible to write a functional definition for the sequence in terms of the first term, 10, and the common difference, 8, and thus:
tn=10+(n-1)×8
=2+8n, n∈{1, 2, 3, …}
We can readily get a general formula for the nth term of an arithmetic sequence whose first term is a and whose common difference is d (see table B).
In general, then:
The nth term of an arithmetic sequence is given by
tn=a+(n-1)×d
=(a–d)+nd, n∈{1, 2, 3, …}
where a is the first term and d is the common difference.
Graphing an arithmetic sequence
● Since an arithmetic sequence involves adding or subtracting the same value repeatedly, the relationship between the terms is a linear one.
● This means that the graph of terms of an arithmetic sequence is a straight line.
WORKED EXAMPLE
For the arithmetic sequence 2, 4, 6, 8, 10, …
(a) Draw up a table showing the term number with its value
(b) Graph the values in the table
(c) from your graph, determine the value of the tenth term in the sequence.
SOLUTION:
(a) Draw up a table to show the term number and value.
Term number
1
2
3
4
5
Term value
2
4
6
8
10
(b) The value of the term depends on the term number, so ‘value’ is graphed on the y-axis, Draw up a suitable scale on both axes, plot the points and join with a straight line. Note that, even though the graph is a straight line, values on the line only have meaning for integer values of the term number.
(c) (1) Find the term number 10 on the x-axis, and draw a vertical line from the x-axis to meet the straight line. Read the y-value of this point.
(2) Write the answer.
The tenth number in this sequence is 20.
Arithmetic Sequence:
A sequence in which each term after the first term is obtained from the preceding term by adding a fixed number, is called as an arithmetic sequence or Arithmetic Progression, it is denoted by A.P.
e.g., (i) 2, 4, 6, 8, 10, 12, …
(ii) 1, 3, 5, 7, 9, 11, …
Common Difference:
The fixed number in above definition is called as common difference. It is denoted by d. it is obtained by subtracting the preceding terms from the next term i.e; an–a(n-1) ; n>1.
For example
2, 4, 6, 8, 4, …
d= Common difference =a2–a1=4-2=2
Or d= Common difference =a3–a2=6-4=2
The General Form of an Arithmetic Progression:
Let ‘a’ be the first term and ‘d’ be the common difference, then General form of an arithmetic progression is
a, a+d, a+2d, …, a+(n-1)d
A sequence is a list of numbers or objects, called terms, in a certain order. In an arithmetic sequence, the difference between one term and the next is always the same. This difference is called a common difference. The common difference is added to each term to get the next term.
2, 5, 8, 11, 14, …,
this is an increasing arithmetic sequence with a common difference of 3.
32, 26, 20, 14, 8, …,
this is a decreasing arithmetic sequence with a common difference of –6.
If asked to extend either of these patterns (the next 3 terms), you need to know the last term and then you can go on from there. These are examples of recursive sequences.
Example: What are the next three terms in the sequence?
1, 5, 9, 13, … I can see that this is an arithmetic sequence with a common difference of 4. To get the next three terms, add 4 to 13 which equals 17, the next term in the sequence. Then add 4 to 17 to get the next term to get 21, etc. So the next three terms are 17, 21, and 25. In a recursive sequence, you need to know the previous term to get the next term.
In an explicit sequence, you can calculate any term in a sequence in a direct way using the first term and the common difference.
Use the following formula to find any term of an arithmetic sequence.
an=a1+(n-1)d
an= the term in the sequence you are trying to find (n represents the desired term number) a1= the first term in the sequence d= the common difference
Important: Arithmetic Sequences
A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant: a2–a1=a3–a2=an–a(n-1) =d (36.3)
This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condition does not hold, the sequence is not an arithmetic sequence.
Extension: Plotting a graph of terms in an arithmetic sequence
Plotting a graph of the terms of sequence sometimes helps in determining the type of sequence involved. For an arithmetic sequence, plotting an vs. n results in:
Given a term, can you write the next term in each of the lists above? If so, how will you write it? Perhaps by following a pattern or rule. Let us observe and write the rule.
In (i), each term is l more than the term preceding it.
In (ii), each term is 30 less than the term preceding it.
In (iii), each term is obtained by adding 1 to the term preceding it.
In (iv), all the terms in the list are 3, i.e., each tenn is obtained by adding (or subtracting) 0 to the term preceding it.
In (v), each term is obtained by adding -0.5 to (i.e., subtracting 0.5 from) the term preceding it.
In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression (AP).
So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.
Let us denote the first term of an AP by a1, second term by a2, …, nth term by an and the common difference by d. Then the AP becomes a1, a2, a3, …, an.
So, a2–a1=a3–a2=d.
Some more examples of AP are:
(a) The heights (in cm ) of some students of a school standing in a queue in the morning assembly are 147 , 148, 149, …, 157.
(b) The minimum temperatures (in degree Celsius ) recorded for a week in the month of January in a city, arranged in ascending order are
-3.1, -3.0, -2.9, -2.8, -2.7, -2.6, -2.5
(c) The balance money (in $) after paying 5% of the total loan of $1000 every month is 950, 900, 850, 800, …, 50.
(d) The cash prizes (in $) given by a school to the toppers of Classes I to XII are, respectively, 200, 250, 300, 350, …, 750.
(e) The total savings (in $) after every month for 10 months when $50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.
It is left as an exercise for you to explain why each of the lists above is an AP.
You can see that
a, a+d, a+2d, a+3d, …
represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.
Note that in examples (a) to (e) above, there are only a finite number of terms. Such an AP is called a finite AP. Also note that each of these Arithmetic Progressions (APs) has a last term. The APs in examples (i) to (v) in this section, are not finite APs and so they are called infinite Arithmetic Progressions. Such APs do not have a last term.
