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 :
25^{th} term : 6, 2, -2, -6,.........
Problem 6 :
16^{th} term : a_{1} = 8; d = 5
Problem 7 :
60^{th} term : 11, 5, -1, -7,.........
Problem 8 :
12^{th} term : a_{1} = 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?
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.
a_{n} = a_{n-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.
a_{n} = a_{n-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 25^{th} term.
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 6 for a_{1}, 25 for n, and -4 for d.
a_{25} = 6 + (25 - 1)(-4)
Simplify the expression in parentheses.
a_{25} = 6 + (24)(-4)
Multiply.
a_{25} = 6 - 96
Subtract.
a_{25} = -90
6. Answer :
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 8 for a_{1}, 16 for n, and 5 for d.
a_{16} = 8 + (16 - 1)(5)
Simplify the expression in parentheses.
a_{16} = 8 + (15)(5)
Multiply.
a_{16} = 8 + 75
Add.
a_{16} = 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 60^{th} term.
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 11 for a_{1}, 60 for n, and -6 for d.
a_{60} = 11 + (60 - 1)(-6)
Simplify the expression in parentheses.
a_{60} = 11 - (59)(-6)
Multiply.
a_{60} = 11 - 354
Subtract.
a_{60} = -343
8. Answer :
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 4.2 for a_{1}, 12 for n, and 1.4 for d.
a_{12} = 4.2 + (12 - 1)(1.4)
Simplify the expression in parentheses.
a_{12} = 4.2 + (11)(1.4)
Multiply.
a_{12} = 4.2 + 15.4
Add.
a_{12} = 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, a_{1} = 60,180. Because you want to find the odometer reading on day 22, you will need to find the 20^{th} term of the sequence, so n = 22.
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 60,180 for a_{1}, 22 for n, and 60 for d.
a_{22} = 60,180 + (22 - 1)(60)
Simplify the expression in parentheses.
a_{22} = 60,180 + (21)(60)
Multiply.
a_{22} = 60,180 + 1,260
Add.
a_{22} = 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, a_{1} = 2000. Because you want to find the weight after stop 6, you will need to find the 6^{th} term of the sequence, so n = 6.
Write the rule to find the n^{th} term.
a_{n} = a_{1} + (n - 1)d
Substitute 2000 for a_{1}, 6 for n, and -250 for d.
a_{6} = 2000 + (6 - 1)(-250)
Simplify the expression in parentheses.
a_{6} = 2000 + (5)(-250)
Multiply.
a_{6} = 2000 - 1250
Subtract.
a_{6} = 750
The load weighs 750 pounds after stop 6.
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