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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
1d,
5
3u1
Your Turn: Write the recursive form and find the next 3 terms
u1 = 8, d = –2 u1 = –9.2, d = 0.9
Writing Arithmetic Sequences 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
Find the recursive formula u2 = –5 and u7 = 30
Example #3
Find the recursive formula u4 = –43 and u6 = –61
Your TurnFind 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
Writing Arithmetic Sequences 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
Writing Arithmetic Sequences 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 #2
u7 = -8 and d = 3
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
Your Turn:
1. u1 = 4, d = ¼ 2. u1 = -6, d = ⅔
Find u8 Find u4
3. u1 = 10, d = -½ 4. u1 = π, d = 2
Find u12 Find u27
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
1nnuSigma
(Summation Symbol)
Upper Bound (Ending Term #)
Lower Bound (Starting Term #)
Sequence
Example #1
4
1nn2
Example #2
3
1n)3n(
Example #3
)2n3(3
1n
Your Turn: Find the sum:
5
1n)7n3(
4
1n)n45(
Your Turn: Find the sum:
5
1n)n37(
4
1n]4)1n(3[
Your Turn: Find the sum:
5
1n
2 )n30(
4
1n)2n(n
Partial Sums of Arithmetic Sequences – Formula #1
Good to use when you know the 1st term AND the last term
k
1nk1n )uu(
2
ku
# of terms
1st term last term
Formula #1 – Example #1Find the partial sum:
k = 9, u1 = 6, u9 = –24
Formula #1 – Example #2Find the partial sum:
k = 6, u1 = – 4, u6 = 14
Formula #1 – Example #3Find the partial sum:
k = 10, u1 = 0, u10 = 30
Your Turn:
Find the partial sum:
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
1n1n d
2
)1k(kkuu
# of terms
1st term common difference
Formula #2 – Example #1Find the partial sum:
k = 12, u1 = –8, d = 5
Formula #2 – Example #2Find the partial sum:
k = 6, u1 = 2, d = 5
Formula #2 – Example #3Find the partial sum:
k = 7, u1 = ¾, d = –½
Your Turn:
Find the partial sum:
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
1n1n d
2
)1k(kkuu
k
1nk1n )uu(
2
ku
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