What are Recurrence Relations?

If there is a relationship between the elements of a sequence, the recurrence relation would be the rule that connects them. It is possible to formulate the recurrence relation and use it to find the value of each of the elements of the set according to the position it occupies.

For example

Recurrence Relations

Ways to Find Recurrence Relations

There are several ways to find recurrence relations. One is to observe the sequence of elements and how they change. Another way is to write down parameters in a table.

A rule can be formulated using addition, subtraction, multiplication or division—or several of these operations together.

Let's look at an example:

Consider the sequence of elements: 3,7,11,15,19 3,7,11,15,19 .

If we look closely at the numbers, we can see that there is a certain rule of formation between them: to get from one number to the next, we need to add 4 4 each time.

The first element is 3 3 . If we add 4 4 to this number, we will get the second element, which is 7 7 . If we add 4 4 to this number again, we will arrive at the third element (11 11 ), and so on.

Therefore the rule in this case is: +4 +4 .


Practice Series

Examples with solutions for Series

Exercise #1

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

Video Solution

Step-by-Step Solution

It is possible to see that there is a difference of one number between each number.

That is, 1 is added to each number and it will be the next number:

1+1=2 1+1=2

2+1=3 2+1=3

3+1=4 3+1=4

Etcetera. Therefore, the next numbers missing in the sequence will be:8+1=9 8+1=9

10+1=11 10+1=11

Answer

11 , 9

Exercise #2

Look at the following set of numbers and determine if there is any property, if so, what is it?

94,96,98,100,102,104 94,96,98,100,102,104

Video Solution

Step-by-Step Solution

One can observe that the difference between each number is 2.

That is, with each leap the next number increases by 2:

94+2=96 94+2=96

96+2=98 96+2=98

98+2=100 98+2=100

and so forth......

Answer

+2 +2

Exercise #3

The table shows the number of balls and the number of courts at the school:

246123BallsCourts

.

Complete:

Number of balls is _________ than the number of courts

Video Solution

Step-by-Step Solution

It is possible to see that if you multiply each number from the right column by 2, you get the number from the left column.

That is:1×2=2 1\times2=2

2×2=4 2\times2=4

3×2=6 3\times2=6

Therefore, the number of balls is 2 times greater than the number of courts.

Answer

2 times greater

Exercise #4

The sequence below is structured according to a term-to-term rule.

What is the first element?

?+? \text{?}+\text{?}

2+4 2+4

3+7 3+7

4+10 4+10

5+13 5+13

Video Solution

Step-by-Step Solution

We start with the right column in the exercises.

Between each number there is a jump of +3:4+3=7 4+3=7

7+3=10 7+3=10

Etcetera.

Now we move to the left column of the exercises.

Between each number there is a jump of +1:

2+1=3 2+1=3

3+1=4 3+1=4

Now we can figure out which exercise is missing:

The left digit will be:21=1 2-1=1

The right digit will be:43=1 4-3=1

And the missing exercise is:1+1 1+1

Answer

1+1 1+1

Exercise #5

Below is a sequence represented by squares. How many squares will there be in the 8th element?

Video Solution

Step-by-Step Solution

It is apparent, that for each successive number, a square is added in length and one in width.

Hence, the rule using the variable n is:

a(n)=n2 a(n)=n^2

Therefore, the eighth term will be:

n2=8×8=16 n^2=8\times8=16

Answer

64 64

Exercise #6

Below is the rule for a sequence written in terms of n n :

2n+2 2n+2

Calculate the value of the 11th element.

Video Solution

Step-by-Step Solution

We calculate by replacingn=11 n=11

2×11+2= 2\times11+2=

First we solve the multiplication exercise and then we add 2:

22+2=24 22+2=24

Answer

24 24

Exercise #7

Given a series whose first element is 15, each element of the series is less by 2 of its predecessor.

Is the number 1 an element of the series?

Video Solution

Step-by-Step Solution

We know that the first term of the series is 15.

From here we can easily write the entire series, until we see if we reach 1.  

15, 13, 11, 9, 7, 5, 3, 1

 

The number 1 is indeed an element of the series!

Answer

Yes

Exercise #8

A sequence has the following term-to-term rule:

n2 \frac{n}{2}

What is the the third term?

Video Solution

Step-by-Step Solution

The third term in the sequence is the term a3 a_3 meaning in the general term formula given:

an=n2 a_n= \frac{n}{2} We need to substitute the position (of the requested term in the sequence):

n=3 n=3 Let's do this:

an=n2n=3a3=32 a_{\underline{n}}= \frac{\underline{n}}{2} \\ n=\underline{3}\\ \downarrow\\ a_{\underline{3}}=\frac{\underline{3}}{2} When we substituted in place of n the position (of the requested term in the sequence): 3, the substitution is shown with an underline in the expression above,

Therefore, the correct answer is answer C.

Answer

32 \frac{3}{2}

Exercise #9

10n9 10n-9

What are the fourth and fifth terms of the sequence above?

Video Solution

Step-by-Step Solution

The fourth and fifth terms in the sequence are the terms: a4,a5 a_4,\hspace{4pt}a_5 meaning in the general term formula given:

an=10n9 a_n=10n-9 we need to substitute the position (of the requested term in the sequence):

n=4 n=4 for - a4 a_4 and-

n=5 n=5 for-

a5 a_5 Let's do this for the fourth term:

an=10n9n=4a4=1049=409a4=31 a_{\underline{n}}= 10\underline{n}-9 \\ n=\underline{4}\\ \downarrow\\ a_{\underline{4}}= 10\cdot\underline{4}-9=40-9\\ a_4=31 when we substituted in place of n the position (of the requested term in the sequence): 4, substitution is shown with an underline in the expression above,

Similarly, for the fifth term, a5 a_5 we get:

a5=1059=509a5=41 a_{\underline{5}}= 10\cdot\underline{5}-9=50-9\\ a_5=41 which means that:

a4=31,a5=41 a_4=31,\hspace{4pt}a_5=41 Therefore the correct answer is answer A.

Answer

31, 41

Exercise #10

an=n+5 a_n=n+5

Is the number 15 a term in the sequence above?

Video Solution

Step-by-Step Solution

Let's check if the number 15 is a member of the sequence defined by the given general term:

an=n+5 a_n=n+5

We will do this in the following way:

We will require first the existence of such a member in the sequence, at some position, meaning we will require that:

an=15 a_n=15

Then, we will solve the equation obtained from this requirement, while remembering that n is the position of the member in the sequence (also called - the index of the member in the sequence), and therefore must be a natural number, meaning a positive whole number, and therefore we will require this as well,

Let's check if these two requirements are fulfilled together:

First, let's solve:

{an=n+5an=1515=n+5 \begin{cases} a_n=n+5\\ a_n=15 \end{cases}\\ \downarrow\\ 15=n+5

where we substituted an a_n in the first equation with its value from the second equation,

We got an equation with one unknown for n, let's solve it in the regular way by moving terms and isolating the unknown and we get:

15=n+5n=515n=10/:(1)n=10 15=n+5 \\ -n=5-15\\ -n=-10 \hspace{8pt} \text{/:}(-1)\\ n=10

where in the last step we divided both sides of the equation by the coefficient of the unknown on the left side,

We got therefore that the requirement that:

an=15 a_n=15

leads to:

n=10 n=10

and this is indeed a natural number, meaning - positive and whole, and therefore we can conclude that in the sequence defined in the problem by the given general term, the number 15 is indeed a member and its position is 10, meaning - in mathematical notation:

a10=15 a_{10}=15

Therefore the correct answer is answer A.

Answer

Yes

Exercise #11

A sequence has a term-to-term rule of an=15n a_n= 15n .

Is the number 30 a term in the sequence?

Video Solution

Step-by-Step Solution

Let's check if the number 30 is a term in the sequence defined by the given general term:

an=15n a_n= 15n ,

We will do this in the following way:

We will require first the existence of such a term in the sequence, at some position, meaning we will require that:

an=30 a_n=30

Then, we will solve the equation obtained from this requirement, while remembering that n is the position of the term in the sequence (also called - the index of the term in the sequence), and therefore must be a natural number, meaning a positive whole number, and therefore we will require this as well,

Let's check if these two requirements are met together:

First, let's solve:

{an=15nan=3030=15n \begin{cases} a_n= 15n \\ a_n=30 \end{cases}\\ \downarrow\\ 30=15n where we substituted an a_n in the first equation with its value from the second equation,

We got an equation with one unknown for n, let's solve it in the regular way by moving terms and isolating the unknown and we get:

30=15n15n=30/:(15)n=2 30=15n \\ -15n=-30 \hspace{8pt} \text{/:}(-15)\\ n=2

where in the last step we divided both sides of the equation by the coefficient of the unknown on the left side,

We got therefore that the requirement that:

an=30 a_n=30

leads to:

n=2 n=2

and this is indeed a natural number, meaning - positive and whole, and therefore we can conclude that in the sequence defined in the problem by the given general term, the number 30 is indeed a term and its position is 2, meaning - in mathematical notation:

a2=30 a_{2}=30

Therefore the correct answer is answer B.

Answer

Yes, it is the second term.

Exercise #12

Given the series, y represents some term of the series

n represents the position of the term in the series

What are the first five members of the series?

an=3n+1 a_n=3n+1

Video Solution

Step-by-Step Solution

Let's find the first five terms in the sequence by substituting their positions in the given general term formula:

an=3n+1 a_n=3n+1

We want to calculate the values of the terms:

a1,a2,a3,a4,a5 a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5

Let's start with the first term in the sequence,

meaning in the given general term formula:

an=3n+1 a_n=3n+1

We need to substitute the position (of the requested term in the sequence),

We want to find the first term so we'll substitute:

n=1 n=1

Let's perform this:

an=3n+1n=1a1=31+1=4 a_{\underline{n}}= 3\underline{n}+1 \\ n=\underline{1}\\ \downarrow\\ a_{\underline{1}}=3\cdot\underline{1}+1=4

When we substituted in place of n the position (of the requested term in the sequence): 1, the substitution is shown with an underline in the expression above,

We'll repeat this action identically for all the requested terms in the sequence, meaning for the second through fifth terms:

a2=32+1=7a3=33+1=10a4=34+1=13a5=35+1=16 a_{\underline{2}}=3\cdot\underline{2}+1=7 \\ a_{\underline{3}}=3\cdot\underline{3}+1=10 \\ a_{\underline{4}}=3\cdot\underline{4}+1=13 \\ a_{\underline{5}}=3\cdot\underline{5}+1=16 \\ Where for the second term a2 a_2 we substituted:n=2 n=2 in the given general term formula:

an=3n+1 a_n=3n+1

For the third term a3 a_3 we substituted:n=3 n=3 and so on identically for the rest of the requested terms,

To summarize, we found that the first five terms:

a1,a2,a3,a4,a5 a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5

in the given sequence, are:

4,7,10,13,16 4,\hspace{4pt}7,\hspace{4pt}10,\hspace{4pt}13,\hspace{4pt}16

Therefore, the correct answer is answer A.

Answer

4,7,10,13,16 4,7,10,13,16

Exercise #13

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

Video Solution

Answer

Yes

Exercise #14

Look at the following set of numbers and determine if there is any property, if so, what is it?

1,2,3,4,5,6 1,2,3,4,5,6

Video Solution

Answer

+1 +1

Exercise #15

Look at the following set of numbers and determine if there is any property, if so, what is it?

10,8,6,4,2 10,8,6,4,2

Video Solution

Answer

2 -2