Notes 3.4
Arithmetic Sequences
I. Sequences and Terms• Sequence: a list of numbers
in a specific order.
1, 3, 4, 7, 10, 16
• Term: each number in a sequence
Sequence
Terms
A sequence is arithmetic if the differences between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
arithmetic sequence
The common difference, d, is 5.
II. Arithmetic Sequences- Definition and Formulas
FYI: Common differences can be negative.
A. Definition
How do I know if it is an arithmetic sequence?
• Look for a common difference between consecutive terms
Ex: 2, 4, 8, 16 . . . Common Difference?
2+2 = 4
4+ 4 = 8
8 + 8 = 16
No. The sequence is NOT Arithmetic
Ex: 48, 45, 42, 39 . . . Common Difference?
48 - 3 = 45
45 - 3 = 42
42 - 3 = 39
Yes. The sequence IS Arithmetic. d = -3
Example
• Find the next three terms in the arithmetic sequence: 2, 5, 8, 11, 14, __, __, __
• 2, 5, 8, 11, 14, 17, 20, 23
• The common difference is?
• 3!!!
B. Important Formulas for Arithmetic Sequence:
• Recursive Formula • Explicit Formula
an = an – 1 + d an = a1 + (n - 1)d
Where:
an is the nth term in the
sequence
a1 is the first term
n is the number of the term
d is the common difference
III. Finding terms given the formula
• Given the following formulas, find the first four terms.
Ex 2:
t1 = 2
tn = tt-1 -6
2, -4, -10, -16
Ex 3:
t1 = -3
tn = tt-1 +4
-3, 1, 5, 9
Ex 1:
t1 = 8
tn = tt-1 +5
8,13, 18, 23
A. Given Recursive
• Ex 1: Given tn = 5n-2, find the first four terms.
3, 8, 13, 18
B. Given Explicit
For the first term,
substitute 1 for n. t1 = 5(1)-2
t1 = 3 Do the same to
find the second,
third, and fourth
term.
t2 = 5(2)-2
t2 = 8
t3 = 5(3)-2
t3 = 13
t4 = 5(4)-2
t4 = 18
• Given tn = -3n+1 find the first four terms.
-2, -5, -8, -11
B. Given Explicit
For the first term,
substitute 1 for n. t1 = -3(1)+1
t1 = -2 Do the same to
find the second,
third, and fourth
term.
t2 = -3(2)+1
t2 = -5
t3 = -3(3)+1
t3 = -8
t4 = -3(4)+1
t4 = -11
Ex 1: Write the explicit and recursive formula for each sequence
2, 4, 6, 8, 10
First term: a1 = 2
difference = 2
an = an – 1 + dan = an – 1 + 2, where
a1 = 2Recursive:
an = a1 + (n-1)d an = 2 + (n-1)2Explicit:
IV. Write the formulas given the sequence.
Ex 2: Write the explicit and recursive formula for each sequence
7, 11, 15, 19…
First term: a1 = 7
difference = 4
an = an – 1 + dan = an – 1 + 4, where
a1 = 7Recursive:
an = a1 + (n-1)d an = 7 + (n-1)4Explicit:
IV. Write the formulas given the sequence.
V. Recursive to Explicit• Given the recursive formula, write the explicit
formula for the sequence.
Ex 1:
a1 = 8
an = an-1 +5
an = a1 + (n-1)d
an = 8 + (n-1)5
Ex 2:
a1 = -5
an = aa-1 +2
an = a1 + (n-1)d
an = -5 + (n-1)2
VI. Explicit to Recursive
an = an – 1 + d
an = an – 1 + 4, where
a1 = 6
Ex 1: an = 4n + 2
a1 = 4(1) + 2
a1 = 6
a2 = 4(2) + 2
a2 = 10
d = 4
an = an – 1 + d
an = an – 1 + 3, where
a1 = -5
Ex 2: an = 3n - 8
a1 = 3(1) - 8
a1 = -5
a2 = 3(2) - 8
a2 = -2
d = 3
VII. Finding the nth term.
Ex 1: Find the 25th term in the sequence of 5, 11, 17, 23, 29 . . .
an = a1 + (n-1)d
a25 = 5 + (24)6 =149
difference = 6
a25 = 5 + (25 -1)6
Start with the explicit sequence formula
Find the common difference
between the values.
Plug in known values
Simplify
Explicit Arithmetic Sequence Problem
Ex 2: Find the 17th term of the arithmetic sequence: 26, 13, 0, -13
an = a1 + (n-1)d
a25 = 26 + (16)(-13) =-182
difference = -13
a25 = 26 + (17 -1)(-13)
Start with the explicit sequence formula
Find the common difference
between the values.
Plug in known values
Simplify
Ex. 3: Suppose you have saved $75 towards the purchase of a new iPad. You plan to save at least $12 from mowing your neighbor’s yard each week. In all, what is the minimum amount of money you will have in 26 weeks?
an = a1 + (n-1)d
a26 = 75 + (26)12 =$387
difference = 12
a26 = 75 + (27 -1)12
Start with the explicit sequence formula
Find the common difference
between the values. You will save
$12 a week so this is your difference.
Plug in known values
Simplify
WAIT: Why 27 and not 26 for n ?
The first term in the sequence, 75, came before the weeks started (think of it
as week 0). Therefore you want one more week in your formula to account
for the $75 that you had before you started saving.