Introduction-to-Arithmetic-Sequences.ppt

Introduction to
Arithmetic Sequences
18 May 2011

Arithmetic Sequences
 When the difference between any two
numbers is the same constant value
 This difference is called d or the constant
difference
 {4, 5, 7, 10, 14, 19, …}
 {7, 11, 15, 19, 23, ...}
← Not an Arithmetic Sequence
← Arithmetic Sequence
d = 4

Your Turn:
 Determine if the following sequences are
arithmetic sequences. If so, find d (the
constant difference).
 {14, 10, 6, 2, –2, …}
 {3, 5, 8, 12, 17, …}
 {33, 27, 21, 16, 11,…}
 {4, 10, 16, 22, 28, …}

Recursive Form
 The recursive form of a sequence tell you the
relationship between any two sequential (in
order) terms.
un = un–1 + d n ≥ 2
common difference

Writing Arithmetic Sequences
in Recursive Form
If given a term and d
1. Substitute d into the recursive formula

Examples: Write the recursive
form and find the next 3 terms
  u1 = 39, d = 5
3
1
d
,
5
3
u1




Your Turn: Write the recursive
form and find the next 3 terms
 u1 = 8, d = –2  u1 = –9.2, d = 0.9

in Recursive Form, cont.
If given two, non-sequential terms
1. Solve for d
d = difference in the value of the terms
difference in the number of terms
2. Substitute d into the recursive formula

Example #1
Find the recursive formula
 u3 = 13 and u7 = 37

Example #2
 u2 = –5 and u7 = 30

Example #3
 u4 = –43 and u6 = –61

Your Turn
Find the recursive formula:
1. u3 = 53 and u5 = 71 2. u2 = -7 and u5 = 8
3. u3 = 1 and u7 = -43

Explicit Form
 The explicit form of a sequence tell you the
relationship between the 1st
term and any
other term.
un = u1 + (n – 1)d n ≥ 1
common difference

Summary: Recursive Form vs.
Explicit Form
Recursive Form
un = un–1 + d n ≥ 2
 Sequential Terms
Explicit Form
un = u1 + (n – 1)d n ≥ 1
 1st
Term and Any
Other Term

in Explicit Form
 You need to know u1 and d!!!
 Substitute the values into the explicit formula
1. u1 = 5 and d = 2 2. u1 = -4 and d = 5

in Explicit Form, cont.
 You may need to solve for u1 and/or d.
1. Solve for d if necessary
2. Back solve for u1 using the explicit formula
u4 = 12 and d = 2

Example #3
u6 = 57 and u10 = 93

Example #4
u2 = -37 and u7 = -22

Your Turn:
Find the explicit formulas:
1. u5 = -2 and d = -6 2. u11 = 118 and d =
13
3. u3 = 17 and u8 = 92 4. u2 = 77 and u5 = -34

Using Explicit Form to Find
Terms
 Just substitute values into the formula!
u1 = 5, d = 2, find u5

Using Explicit Form to Find
Terms, cont.
u1 = -4, d = 5, find u10

Introduction-to-Arithmetic-Sequences.ppt

Summations
 Summation – the sum of the terms in a
sequence
{2, 4, 6, 8} → 2 + 4 + 6 + 8 = 20
 Represented by a capital Sigma


Summation Notation


k
1
n
n
u
Sigma
(Summation
Symbol)
Upper Bound
(Ending Term #)
Lower Bound
(Starting Term #)
Sequence

Example #2



3
1
n
)
3
n
(

Example #3
)
2
n
3
(
3
1
n




Your Turn: Find the sum:



5
1
n
)
7
n
3
( 


4
1
n
)
n
4
5
(





5
1
n
)
n
3
7
( 




4
1
n
]
4
)
1
n
(
3
[




5
1
n
2
)
n
30
( 


4
1
n
)
2
n
(
n

Partial Sums of Arithmetic
Sequences – Formula #1
 Good to use when you know the 1st
term AND the
last term




k
1
n
k
1
n )
u
u
(
2
k
u
# of terms
1st
term last term

Formula #1 – Example #1
Find the partial sum:
k = 9, u1 = 6, u9 = –24

k = 6, u1 = – 4, u6 = 14

k = 10, u1 = 0, u10 = 30

Your Turn:
1. k = 8, u1 = 7, u8 = 42
2. k = 5, u1 = –21, u5 = 11
3. k = 6, u1 = 16, u6 = –19

Partial Sums of Arithmetic
Sequences – Formula #2
 Good to use when you know the 1st
term, the # of
terms AND the common difference





k
1
n
1
n d
2
)
1
k
(
k
ku
u
# of terms
1st
term common difference

k = 12, u1 = –8, d = 5

k = 6, u1 = 2, d = 5

k = 7, u1 = ¾, d = –½

Your Turn:
1. k = 4, u1 = 39, d = 10
2. k = 5, u1 = 22, d = 6
3. k = 7, u1 = 6, d = 5

Choosing the Right Partial
Sum Formula
Do you have the last term or the constant difference?





k
1
n
1
n d
2
)
1
k
(
k
ku
u




k
1
n
k
1
n )
u
u
(
2
k
u

Examples
 Identify the correct partial sum formula:
1. k = 6, u1 = 10, d = –3
2. k = 12, u1 = 4, u12 = 100

Your Turn:
 Identify the correct partial sum formula
and solve for the partial sum
1. k = 11, u1 = 10, d = 2
2. k = 10, u1 = 4, u10 = 22
3. k = 16, u1 = 20, d = 7
4. k = 15, u1 = 20, d = 10
5. k = 13, u1 = –18, u13 = –102

Introduction-to-Arithmetic-Sequences.ppt

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