ARITHMETIC SEQUENCES WORKSHEET

Problem 1-4 : Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence.

Problem 1 : 

15, 10, 5, 0, ........

Problem 2 : 

2, 5, 10, 17, ........

Problem 3 : 

-3/4, -1/4, 1/4, 3/4, ........

Problem 4 : 

-4, -2, 1, 5, ........

Problems 5-8 : Find the indicated term of each arithmetic sequence.

Problem 5 :

25th term : 6, 2, -2, -6,.........

Problem 6 :

16th term : a1 = 8; d = 5

Problem 7 :

60th term : 11, 5, -1, -7,.........

Problem 8 :

12th term : a1 = 4.2; d = 1.4

Problem 9 :

The odometer on a car reads 60,180 on day 1. Every day, the car is driven 60 miles. If this pattern continues, what is the odometer reading on day 22?

Problem 10 :

Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6?

Detailed Answer Key

1. Answer : 

Step 1 :

15, 10, 5, 0, ........

Find the difference between successive terms.

10 - 15  =  -5

5 - 10  =  -5

0 - 5  =  -5

The common difference is -5.

Step 2 :

Use the common difference to find the next 3 terms.

Add -5 to each term to find the next term.

an  =  an-1 + d

0 + (-5)  =  0 - 5  =  -5

-5 + (-5)  =  -5 - 5  =  -10

-10 + (-5)  =  -10 - 5  =  -15

The sequence appears to be an arithmetic sequence with a common difference of -5.

The next 3 terms are -5, -10, -15.

2. Answer : 

2, 5, 10, 17, ........

Find the difference between successive terms.

5 - 2  =  3

10 - 5  =  5

17 - 10  =  7

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

3. Answer : 

Step 1 :

-3/4, -1/4, 1/4, 3/4, ........

Find the difference between successive terms.

-1/4 - (-3/4)  =  1/2

1/4 - (-1/4)  =  1/2

3/4 - 1/4  =  1/2

The common difference is 1/2.

Step 2 :

Use the common difference to find the next 3 terms.

Add 1/2 to each term to find the next term.

an  =  an-1 + d

3/4 + 1/2  =  3/4 + 2/4  =  5/4

5/4 + 1/2  =  5/4 + 2/4  =  7/4

7/4 + 1/2  =  7/4 + 2/4  =  9/4  

The sequence appears to be an arithmetic sequence with a common difference of 1/2.

The next 3 terms are 5/4, 7/4, 9/4.

4. Answer : 

-4, -2, 1, 5, ........

Find the difference between successive terms.

-2 - (-4)  =  2

1 - (-2)  =  3

5 - 1  =  4

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

5. Answer : 

Step 1 : Find the common difference.

2 - 6  =  -4

-2 - 2  =  -4

-6 - (-2)  =  -4

The common difference is -4.

Step 2 : Find the 25th term.

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 6 for a1, 25 for n, and -4 for d.

a25  =  6 + (25 - 1)(-4)

Simplify the expression in parentheses.

a25  =  6 + (24)(-4)

Multiply. 

a25  =  6 - 96

Subtract. 

a25  =  -90

6. Answer : 

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 8 for a1, 16 for n, and 5 for d.

a16  =  8 + (16 - 1)(5)

Simplify the expression in parentheses.

a16  =  8 + (15)(5)

Multiply. 

a16  =  8 + 75

Add. 

a16  =  83

7. Answer : 

Step 1 : Find the common difference.

5 - 11  =  -6

-1 - 5  =  -6

-7 - (-1)  =  -6

The common difference is -6.

Step 2 : Find the 60th term.

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 11 for a1, 60 for n, and -6 for d.

a60  =  11 + (60 - 1)(-6)

Simplify the expression in parentheses.

a60  =  11 - (59)(-6)

Multiply. 

a60  =  11 - 354

Subtract. 

a60  =  -343

8. Answer : 

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 4.2 for a1, 12 for n, and 1.4 for d.

a12  =  4.2 + (12 - 1)(1.4)

Simplify the expression in parentheses.

a12  =  4.2 + (11)(1.4)

Multiply. 

a12  =  4.2 + 15.4

Add. 

a12  =  19.6

9. Answer : 

Notice that the sequence for the situation is arithmetic with d = 60 because the odometer reading will increase by 60 miles per day. Since the odometer reading on day 1 is 60,180 miles, a1 = 60,180. Because you want to find the odometer reading on day 22, you will need to find the 20th term of the sequence, so n = 22.

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 60,180 for a1, 22 for n, and 60 for d.

a22  =  60,180 + (22 - 1)(60)

Simplify the expression in parentheses.

a22  =  60,180 + (21)(60)

Multiply. 

a22  =  60,180 + 1,260

Add. 

a22  =  61,440

The odometer will read 61,440 miles on day 22.

10. Answer : 

Notice that the sequence for the situation is arithmetic with d = -250, because the truck drops off 250 pounds each time it stops. Because cargo weighed 2000 pounds after stop 1, a1 = 2000. Because you want to find the weight after stop 6, you will need to find the 6th term of the sequence, so n = 6.

Write the rule to find the nth term.  

an  =  a1 + (n - 1)d

Substitute 2000 for a1, 6 for n, and -250 for d.

a6  =  2000 + (6 - 1)(-250)

Simplify the expression in parentheses.

a6  =  2000 + (5)(-250)

Multiply. 

a6  =  2000 - 1250

Subtract. 

a6  =  750

The load weighs 750 pounds after stop 6. 

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