Arithmetic sequences and applications.

1. Next in sequence? by flickr user
the mad LOLscientist

2. Some Definitions
Sequence: An ordered list of numbers that follow a certain pattern
(or rule).
Arithmetic Sequence: (i) Recursive Definition: An ordered list of
numbers generated by continuously adding a value (the
common difference) to a given first term.
(ii) Implicit Definition: An ordered list of numbers where
each number in the list is generated by a linear equation.
Common Difference (d): (i) The number that is repeatedly added to
successive terms in an arithmetic sequence.
(ii) From the implicit definition, d is the slope of the linear
equation.

3. To Find The Common Difference
d is the common difference d = tn - t(n - 1)
tn is an arbitrary term in the sequence
t(n - 1) is the term immediately before tn in the sequence
To Find the nth Term In an Arithmetic Sequence
tn is the nth term tn = a + (n - 1)d
a is the first term
n is the quot;rankquot; of the nth term in the sequence
d is the common difference
Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ...
Implicitly
Solution: a = 11 t51 = 11 + (51 - 1)(-6)
d = 5 - 11 t51 = 11 + (50)(-6)
= -6 t51 = 11 - 300
n = 51 t51 = -289

4. Which of the following sequences are arithmetic sequences?
a) 1, 2, 6, 24, 120,…
HOMEWORK
b) 3, 9, 15, …
c) 2, 4, 8, 16, 32,…
d) 1, 2, 3, 5, 8, 13, …
e) -4, -1, 2, 5, 8,…

5. What is the pattern in the sequence 2, 8, 14, 20, 26…?
Suggest an equation that could be used to generate such a list.
HOMEWORK
2, 8, 14, 20, 26

6. a) Why do the numbers 5, 8, 11, 14, 17… form an arithmetic
sequence?
HOMEWORK
b) What is the defining equation that produced them?
c) What is the 27th term of this sequence?
5, 8, 11, 14, 17

7. a) What is the next term of the sequence 1, -1, -3, -5, -7,…?
b) Find an equation that could be used to generate such a sequence.
c) What is the 35th term of this sequence?

8. Harriet wants to phone her aunt in England to give her some family
news. The first minute of the phone call costs $3.60 and each
additional minute costs $0.12.
Set up an equation that could be used to define this relationship.
Use the equation to calculate how much it would cost to talk for 28
minutes.

9. List the first 4 terms of the sequence determined by each of the
following implicit definitions.
(a)
(b)
(c)
(d)

10. Determine which of the following sequences are arithmetic. If a
sequence is arithmetic, write the values of a and d.
(a) 5, 9, 13, 17, ...
(b) 1, 6, 10, 15, 19, ...
(c)-1, -4, -7, -10, ...
(d) x, 2x, 3x, 4x, ...

11. Given the values of a and d, write the first 5 terms of each arithmetic
sequence.
(a) a = 7, d, = 2
(b) a = -4, d, = 6
(c) a = 8, d, = x
(d) a = 3m, d, = 1 - m

12. Find the indicated terms for each arithmetic sequence.
(a) 6, 8, 10, ... t10 and t4
(b) 9, 16, 23, ... t18 and t41
(c) -4, -9, -14, ... t18 and t66
(d) x, x + 4, x + 8, ... t 14 and t n

13. Find the number of terms in each of the following arithmetic sequences.
(a) 10, 15, 20, ..., 250
(b) 40, 38, 36, ...-30
(c) -2, -8, -14, ..., -206
(d) x + 2, x + 9, x + 16, ... , x + 303

14. Complete each arithmetic sequence by finding the missing arithmetic
means.
(d) -1.5, ____, ____, ____, 4.5
(a) 1, ____, 25
(e) 2, ____, ____, ____, ____, 107
(b) 14, ____, ____, 32
(f) m + 40, ____, ____, ____, m + 4
(c) -3, ____, ____, -60

15. The eighth term of an arithmetic sequence is 5.3 and the fourteenth
term is 8.3. What is the fifth term?

16. Answers to the last 7 questions:
1. (a) 0, 3, 6, 9 (b) 0, 1, 4, 9 (c) 1, 2, 4, 8 (d)
(c) -1, -3 (d) x, x
2. (a) 5, 4 (b) not arithmetic
3. (a) 7, 9, 11, 13, 15 (b) -4, 2, 8, 14, 20
(c) 8, 8 + x, 8 + 2x, 8 + 3x, 8 + 4x (d) 3m, 2m + 1, m + 2, 3, 4 - m
(d) x + 4(n - 1), x + 52
4. (a) 24, 72 (b) 65, 702 (c) -89, -329
5. (a) 49 (b) 36 (c) 35 (d) 44
6. (a) 13 (b) 20, 26 (c) -22, -41
(f) m + 31, m + 22, m + 13
(d) 0, 1.5, 3 (e) 23, 44, 65, 86
7. 3.8