Arithmetic Sequence: Recursive & Explicit Formula 

Connect Sequences and Function

Example1

a. Is the Ordered List 26, 39, 52, 65, 78 an Arithmetic Sequence?

Solution:

A sequence is an ordered list of numbers that often forms a pattern. Each number is a term of the

sequence. In an arithmetic sequence, the difference between any two consecutive terms is a constant

called the common difference.

Find the differences between pairs of consecutive terms.

There is a common difference of 13. So, this is an arithmetic sequence.

b. How Are the Sequences Related to Functions?

You can think of a sequence as a function where the domain is restricted to the natural numbers and the

The range is the term of the sequence.

For the sequence 26,39,52,65,78

Let n = the term number in the sequence.

Let A(n) = the value of the nth term of the sequence.

The 1^st term is 26.            A (1) =26                                                                       

                                         A (2) =39                     The 2^nd term is 39.

c. How Do You Represent Sequences Using Subscript Notation?

The subscript notation is commonly used to describe sequences.

a₂=39.                The 2nd term is 39.

You can use either function or subscript notation to represent sequences.

Try It!

1. Is the domain of the function in Part B of Example 1 continuous-discrete? Explain.

Solution:

The domain is discrete because it consists of whole numbers or integers.

Recursive Formula

Recursive, in mathematics, means to repeat a process over and over again, using the output of each step as the next input. A recursive formula relates each term of a sequence to the previous term. It is composed of an initial value and a rule for generating the sequence. The recursive formula for an arithmetic sequence is:

A recursive formula describes the pattern of a sequence and can be used to find the next term in a sequence.

Apply the Recursive Formula

Example 2

1. What is a recursive formula for the height above the ground of the nth step of the pyramid shown?

Use the recursive formula.

The formula an a_n = a_n-1 + 26 gives the height above the ground of the n^th step with a₁ = 26.

2. Use the Recursive Formula to Find the Height above the Ground of the 3rd Step.

Find the height above the ground of the 2nd step.

a_{1 =26}

a₂ = a₁ + 26             

a₂ = 26 + 26 =52                      Use a_{1  ,} to find a₂ .

Find the height above the ground of the 3rd step. 

a₂ = 52

a₃ =a_{2 +} 26                         Use a₂ to find a₃

a_{3 =} 52 +26 = 78

The 3rd step is 78 cm above the ground.

Try It!

• Write a recursive formula to represent the total height of the nth stair above the ground if the height of each stair is 18 cm.

Solution:

Use the recursive formula of an arithmetic sequence given by a_n = a_n-1 + where d is the common difference.

Each step of the stair is 18 cm so d = 18.

nth term: a_n =  a_{n-1 + 18}

Explicit Formula

Definition

An explicit formula expresses the nth term of a sequence in terms of n.

The explicit formula for an arithmetic sequence is:

Apply the Explicit Formula

Example 3

a. The cost of renting a bicycle is given in the table. How can you represent the rental cost using an explicit formula?

b. What is the cost of renting the bicycle for 10 days?

c. How is the explicit formula of an arithmetic sequence related to a linear function

Solution:

To find the rental cost for n days, write an explicit formula for the nth term of the sequence.

Use the explicit formula.

a _n = a₁ + (n-1) d

a _n = 26 + (n-1).12 ………………Substitute 26 for a₁ and 12 for d.  

= 26+ 12n – 12 ………………. Distributive Property     

= 14 + 12n ……………………… Simplify.

The explicit formula a_n= 14 + 12n gives the rental cost for n days.

b.Solution:

Use the explicit formula to find the 10th term in the sequence.

Use the explicit formula to find the 10th term in the sequence

a_n = 14 + 12n

a₁₀ = 14 + 12(10) ……………………Substitute 10 for n.      

a₁₀ = 134

The 10th term in a sequence is 134.

 It costs $134 to rent the bicycle for 10 days.

c.Solution:

The formula a, = 14 + 12n shows that the cost, an, is a function of the number of days, n, the bicycle is rented.

You can write this as a linear function, f(x)= 12x + 14,

or as an equation in slope-intercept form, y = 12x + 14.

The common difference, 12, corresponds to the slope of the graph

COMMON ERROR You might think that the initial value of the sequence and the y-intercept of the graph of the equivalent linear function are the same, but they will not be the same if the initial value of the sequence is not zero.

Try It!

3. The cost to rent a bike is $28 for the first day plus $2 for each day after that. Write an explicit formula for the rental cost for n days. What is the cost of renting the bike for 8 days?

Solution:

Use the explicit formula of an arithmetic sequence given by where a₁ is the first term and d is the common difference. a_n = a₁ + (n-1) d

The cost on the first day is $28 so a_{1 =} 28

and increases by $2 for each day after that so d = 2

a_n = a₁ +(n-1) d

=28 + (n-1)2

=28+2n-2

=a_{n =} 2n+28

a₈₌2 x 8 +28

= 16 +28

a₈=$42 The cost of renting the bike for 8 days=$42

Write an Explicit Formula from a Recursive Formula

Example 4

The recursive formula for the height above the ground of the nth step of the stairs shown is a_n = a_n-1  + 4 with a₁ = 7 What explicit formula finds the height above the ground of the nth step?

Solution:

Use the recursive formula to find information about the sequence.

a₁ = 7

a_n =a_n-1 + 4 common difference

Write the explicit formula.

a_n = a₁ + (n-1) d

a_{n =} 7 + (n-1)4 ………………. substitute 7 for a1 and 4 for d

The explicit formula a_n = 7+ (n-1)4 can be used to find the height above the ground of the nth step.

a_n = 7 + (n-1) 4

= 7+ 4n-4

a_n=4n+3

Try It!

• Write an explicit formula for each arithmetic sequence.
• a. a_n = a _n-1 – 3; a_{1 =} 10                                      b.a_n = a _n-1   + 2.4; a₁ = -1

Solution:

The explicit formula of an arithmetic sequence is given where a₁ is the first term and d is a common difference. a_n = a₁ + (n-1) d

1. From the given, a_n = 10 and d = -3

so, the explicit formula is: a_n =10+ (n − 1) (-3)

=10-3n+3

= -3n+13

b. From the given, a₁= −1 and d = 2.4

so, the explicit formula is: a_n= −1 + (n − 1) (2.4)

= −1+2.4n – 2.

a_n=2.4n – 3.4

Write a Recursive Formula from an Explicit Formula

Example 5

The explicit formula for an arithmetic sequence is a_n= 1 + 1/2 n. What is the recursive formula for the sequence?

Solution:

Step 1: Identify the common difference.

a_n= 1 + 1/2 n.

 d =1/2

Step 2: 

Find the first term of the sequence.

a_n = 1 + ½ n

a₁= 1 + ½ (1)    ………………………..Substitute 1 for  n

a_n = 3/2                  …………………..Simplify.

Step 3:   

Write the recursive formula.

a_n = a_n-1+ d

a_n = a_n-1+1/2             ……………………substitute  ½ for d

The recursive formula for the sequence is :

First term a₁=  3/2 ;

nth term a_n = a_n-1+ 1/2           

Try It!

• Write a recursive formula for each explicit formula.
• a. a_n = 8 +3n                                  b. a_n = 12 – 5n

a. Solution:

Use the recursive formula of an arithmetic sequence given by: a_n = a₁ + (n-1) d where d is a common difference.

Given the explicit formula, we can easily identify the common difference, which is the coefficient of the variable, n.

Here, we have d = 3 Next, we identify the first term by substituting n = 1 to the recursive formula:           

a₁ = 8 + 3(1)

a₁ = 11

When writing a recursive formula for an arithmetic sequence, including the value of the first term, a₁:     

first term: a₁ =11; nth term: a_n= a_n-1 + 3

b.Solution:

Use the recursive formula of an arithmetic sequence given by: a_n = a₁ + (n-1) d where d is a common difference.

Given the explicit formula, we can easily identify the common difference, which is the coefficient of the variable, n.

 Here, we have d = -5 Next, we identify the first term by substituting n = 1 to the recursive formula:

a₁ = 12 – 5(1)

a₁ = 7

When writing a recursive formula for an arithmetic sequence, including the value of the first term, a1:

first term: a₁ = 7; nth term: a_n= a_n-1 – 5

Make Sense and Persevere

1. The lowest and leftmost note on a piano keyboard is an A. The next lowest A is seven white keys to the right. This pattern continues. Write an explicit formula for an arithmetic sequence to represent the position of each A key on the piano, counting from the left. If a piano has 52 white keys, in what position is the key that plays the highest A?

Solution:

Use the explicit formula of an arithmetic sequence given by:

a_n = a₁ + (n − 1)d – where a₁ is the first term and d is a common difference.

The leftmost A corresponds to a₁ = 1 and since the A’s are 7 keys apart, the d = 7:

a_n = 1 + (n − 1)(7)

= 1+7n–7

a_n = 7n – 6

Writing out the A keys that are less than 52, we have: 

Hence, the highest A is at the 50th key.                                  

a₁= 1

a₂=7(2) – 6=8

a₃= 7(3) – 6 = 15

a₄= 7(4) – 6 = 22

a₅= 7(5) – 6 = 29

a₆= 7(6) – 6 = 36

a₇= 7(7) – 6 = 43

a₈= 7(8) – 6 = 50

Hence, the highest A is at the 50^th key.

Check your knowledge

1. Make Sense and Persevere: After the first raffle drawing, 497 tickets remain. After the second raffle drawing, 494 tickets remain. Assuming that the pattern continues, write an explicit formula for an arithmetic sequence to represent the number of raffle tickets that remain after each drawing. How many tickets remain in the bag after the seventh raffle drawing?
2. In a video game, you must score 5,500 points to complete level 1. To move through each additional level, you must score an additional 3,250 points. What number would you use as a₁ when writing an arithmetic sequence to represent this situation? What would n represent? Write an explicit formula to represent this situation. Write a recursive formula to represent this situation.

3. Tell whether each sequence is an arithmetic sequence or not.

4. 4, 7, 10, 14,…

• Write a recursive formula for each sequence.

81, 85, 89, 93, 97,……………

Check Your Knowledge-Answers

Use the explicit formula of an arithmetic sequence given by: a_n= a₁+ (n − 1) d where a₁ is the first term and d is a common difference.

Based on the sequence, the common difference is d = 494 – 497 = -3.

Given that 497 tickets remain after the first raffle drawing, then a₁ = 497.

Hence, the explicit formula is:

a_n= 497 + (n − 1) (−3)

= 497-3n+3

a_n = 500 – 3n

Substitute n=7 representing the seventh raffle drawing

a_n= 500 – 3(7)

= 500 – 21

a_n = 479

So, 479 tickets remain after the seventh raffle drawing.

Solution 2:

We would use a₁ = 5500, the required points to complete level 1 because succeeding levels require 3250 more points than the previous level, which is the common difference.

n would represent the level (level 1, 2, 3, etc.) in this case.

Use the explicit formula of an arithmetic sequence given by:

a_n= a₁+ (n − 1) d

where a₁ is the first term, and d is the common difference.

So, the explicit formula is: a_n= 5500+ (n – 1).3250

= 5500 + 3250n – 3250

a_n= 3250n + 2250

The recursive formula of an arithmetic sequence given by: an = an-1 + d where d is the common difference.

Writing together with the first term, the recursive formula is:

first term: a₁ = 5500; nth term: a_n= a_n-1 + 3250

Solution 3:

Given series; 4,7,10,14 ,…………..

Check the difference: 7-4=3             

10-7=3

14-10=4

So, the difference between any two consecutive terms is not constant. So, the series is not an arithmetic sequence

Solution 4:

Given series: 81,85,89,93,97.

First find the common difference: 85 – 81 = 4

89 – 85= 4……..                                                           

So, the common difference is d = 4

ecursive formula for an arithmetic sequence is

a_{n =} a_n-1+ d. where is nth term, d is the common difference.

a_{n =} a_n-1 + 4

Exercise

Tell whether each sequence is an arithmetic sequence or not.

1. 4,10,16, 22,…………
2. -2,2, -2,2,-2,………….
3. 1,1,1,2,2,2,3,3,3………….
4. Write a recursive formula for each explicit formula and find the first term of the sequence.
   • a_n = 12 +4n
   • a_n = 102-n
5. Write an explicit formula for each recursive formula.
   • a_{n =} a_n-1 + 17 and a₁ = 7
   • a_{n =} a_n-1 + 5 and a₁ = 4
6. Write a recursive formula for each explicit formula and find the first term of the sequence.
   • a_n = 12 + 4n
   • a_n = 102 -n

Concept Summary

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers that follows a pattern. The difference between two consecutive terms is a constant called the common difference.

Recursive Formula

Used to describe a sequence and find the next few terms

a_{n =} a_n-1+ d

a_n=nth term of the sequence

a_n-1=previous term of the sequence

d=common difference

Explicit Formula

Used to find a specific term in the sequence

a_n = a₁ + (n-1) d

a_n =nth term of the sequence

a₁=first term of the sequence

d=common difference

Numbers

1, 7, 13, 19, 25,……………..

Use the recursive formula to describe the sequence and find the next two terms.

d=7-1=6

a_{n =} a_n-1 + 6

a_{6 =} a_{5 +} 6

= 25+6

a₆ = 31

a₇ = a₆ + 6

= 31+6

a_{7 =} 37

The next two terms are 31 and 37    

Use the explicit formula to find the 15th term in the sequence.

a_n = 1+ (n-1) 6

a₁₅ =1 + (14)6

a₁₅ = 85

Concept Map: