Arithmetic SequencesArithmetic Sequences
Lesson 1.3Lesson 1.3
Arithmetic SequenceArithmetic Sequence
• This is a sequence in which the difference between each term and the preceding term is always constant.
• {10, 7, 4, 1, -2, -5,…}• Is {2, 4, 7, 11, 16,…} an arithmetic
sequence?• Recursive Form of arithmetic sequence
un = un-1 + dFor some constant d and all n ≥ 2
ExampleExample
If {un} is an arithmetic sequence with u1 = 2.5 and u2 = 6 as its first two terms
a. Find the common difference
b. Write the sequence as a recursive
function
c. Give the first six terms of the sequence
d. Graph the sequence
Explicit Form of Arithmetic SequenceExplicit Form of Arithmetic Sequence
• In an arithmetic sequence {un} with common difference d, un = u1 + (n-1)d for every n ≥ 1.
• If u1 = -5 and d = 3 we can find the explicit form by, un = -5 + (n-1)3 = -5 + 3n – 3… leaving us with the explicit form of 3n - 8
ExampleExample• If we wanted to know the 38th term of the
arithmetic sequence whose first three terms are 15, 10, and 5, how would we do that?
Here’s HowHere’s How un = u1 + (n-1)d
= 15 + (38-1)(-5)
= 15 + (-5)(37)
= 15 + -185
= -170
• Lets look at example 6 on page 24 because it is far too exhaustive to write down!
Summation NotationSummation Notation
1 2 31
...m
k mk
c means c c c c
5
1
( 7 3 )n
n
What is the sum of this sequence?
Graphing Calculator ExplorationGraphing Calculator Exploration
• We are going to use the sum sequence key on our graphing calculators
• Find the sum of this little diddy
100
1
5 6n
n
Partial Sums of Arithmetic SequencesPartial Sums of Arithmetic Sequences
• If {un} is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by using either of the following formulas
11
11
1.2
( 1)2.
2
k
n kn
k
nn
ku u u
k ku ku d
There is a proof on this on page 27…if anybody really cares
ExampleExample
• Find the 14th partial sum of the arithmetic sequence 21, 15, 9, 3,…
U14 = u1 + (14 – 1)(-6)
= 21 + (13)(-6)
= 21 + (-78)
= -57
12
1
12(21 57)2
7( 36)
252
nn
u
Find the Sum of all multiples of 4 Find the Sum of all multiples of 4 from 4 to 404!from 4 to 404!
• We know that we are adding 4 + 8 + …, so 4x1, 4x2, 4x3, … and we can take 404 ÷ 4 to get the 101 term.
• What this means is there is 101 multiples of 4 in between 4 and 404
• u1=4, k=101, and u101 = 404! Use form 1
101
1
101 101(4 404) (408) 20,604
2 2nn
u
Here is a little story about LarryHere is a little story about Larry• Larry owns an automobile dealership. He spends $18,000
on advertising during the first year, and he plans to increase his advertising expenditures by $1400 in each subsequent year. How much will Larry spend on advertising during the first 9 years?
9
9
9
1
$18,000 (9 1)(1400)
18,000 1400(8)
18,000 11,200
$29,200
:
9 9($18,000 $29,200) ($47,200)2 2
$212,400
nn
u
u
Next
u
Now…Get To Work Slackers!Now…Get To Work Slackers!