Download Arithmetic Sequences

Presented By Mr. Laws
Algebra 1
 Write arithmetic and geometric sequences both
recursively and with an explicit formula, use them to
model situations, and translate between the two
forms.
 How do I express terms within a geometric sequences
using the common ratio, recursive and explicit
definition?
 When identifying number patterns, you must find the
rule of the pattern.
 The rule of pattern for geometric sequences have
operations that requires multiplication or division
within the number sequence.
 What is rule for the following number patterns?
2, 6, 18, 54, 162…
Rule: x 3
1,000, 200, 40, 8…
Rule: 1/5 0r .20
 Notations such a1, a2, a3, …an is used to represent the
terms of a sequence. The subscripts identify the
position of the terms in the sequence.
Example:
a1 – represents the 1st term in the sequence.
a2– represents the 2nd term in the sequence.
a3 – represents the 3rd term in the sequence.
an – represents any term in the sequence.
 What is a geometric sequence?
 It is a sequence of numbers where the ratio of
consecutive terms is constant This is called the common
ratio (r).
 Example:
 3, 6, 12, 24, 48, … This is an increasing geometric
sequence with a common ration of 2 or 2 times a
number.
 100, 25, 6.25, 1.5625, …This is a decreasing geometric
sequence with a common ration of ¼ or .25 times a
number.
 Recursive definition describes a sequence whose
terms are defined by one or more preceding terms.
 Use the following formula for finding the next terms in
a geometric sequence.
an = (an-1) r
 What are the next three terms (a5 , a6 , a7 ) of the following
sequence?
 {4, 20, 100, 500…}
 Use the formula: an = (an-1) r where r = 5
a1 = 4
an = (an – 1) 5
 a5 = (a5 – 1 ) 5 = a4 (5)= 500 (5) = 2, 500 (5th term)
 a6 = (a6 – 1 ) 5 = a5 (5) = 2, 500(5) = 12,500 (6th term)
 a7 = (a7 – 1 )5 = a6 (5) = 12, 500 (5) = 62, 500 (7th term)
 The next three terms are 2500, 12500, and 62,500.
 The explicit definition allows you to calculate any term in a
sequence in a direct way using the first term and the
common ratio (r) between terms.
 Often numbers can get so large that you may have to use
scientific notation rounded to the nearest tenth. For
example. 12.84300 = 12.8 x 106
 The explicit definition is good for solving real world
problems.
 Use the following formula: an = a1 r(n-1)
 What are the 10th, 25th, and 50th terms of the following
sequence?
 {4, 20, 100, 500…} r=5
Use the formula: an = a1 r(n-1)
 a10 = 4 (5) (10-1) = 4 (5)9 = 7.8 x 106 (10th term)
 a25 = 4 (5) (25-1) = 4 (5)24 = 2.4 x 1017 (25th term)
 a50 = = 4 (5) (50-1) = 4 (5)49 = 7.1 x 1034 (50th term)
 Find the geometric sequence of the following pattern.
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Express terms in scientific notation rounded to the
nearest tenth.
(1, 6, 36, 216, 1,296…) r = ?
Find the 12th, 18th, and 24th term of this geometric
sequence.
a12 = 1 (6) (12-1) = 1 (6)11 = 3.6 x 108 ( 12th Term)
a18 = 1 (6) (18-1) = 1 (6)17 = 1.7 x 1013 ( 18th Term)
a24 = 1 (6) (24-1) = 1 (6)23 = 7.9 x 1017 ( 24th Term)
 Ralph’s new job has a starting salary of $20,000. Find
his salary during his fourth year on the job if he
receives annual raises of 5%.
 a1 = $20,000, r = 1.05, n = 4
 Use the formula: an
= a1 r(n-1)
 a4 = 20,000(1.05)(4-1) = 20,000(1.05)(3) = $23,152.50
 What are patterns?
 What is the meaning of Geometric Sequence?
 What is the common ratio?
 What is the recursive definition?
 What is the explicit definition and how do we use it to
solve real world problems?