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partial sums and infinite series - Miss Sahady PSHS Class...
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Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook 1 March 06, 2014 Nov 186:33 PM Warm-Up: 1. 2. Given the sequence 4,8,16,32,..., answer the following questions. What are the next 3 terms in the sequence? What is the common ratio? What is the 15 th term in the sequence? What term number is equal to 512? 3. 4. Nov 186:28 PM partial sums and infinite series Ch. 12 level 2
Transcript of partial sums and infinite series - Miss Sahady PSHS Class...
Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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March 06, 2014
Nov 186:33 PM
Warm-Up:
1.
2.
Given the sequence 4,8,16,32,..., answer the following questions.
What are the next 3 terms in the sequence?
What is the common ratio?
What is the 15th term in the sequence?
What term number is equal to 512?
3.4.
Nov 186:28 PM
partial sums and infinite
seriesCh. 12 level 2
Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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Target
Agenda
Purpose
Evaluation
Agenda
Purpose
Evaluation
TSWBAT: define an arithmetic and geometric partial and infinite sums
Warm-Up/Homework Check
Lesson
BAT: Extend knowledge of patterns and coming to conclusion about them. Prep for summations and Calculus and Statistics. And understand mathematical induction.
Some exercises
Nov 186:42 PM
Partial Sumscreate a sequence that adds the prior terms together up until the term at that number
s1=
s2=
s3=
s4 = sn=
1st partial sum
2nd partial sum
3rdpartial sum
4th partial sum
nth partial sum
a1
What is the 75th term?
a1 + a2
a1 + a2 + a3a1 + a2 + a3 + a4
a1 + a2 + a3 + a4 + ...+an
... ...
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Sequences of Partial Sums:an = 1/2n
s1=
s2=
s3=
s4 = sn=
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ex. ak= k2
S4=
ex. ak= k
S6=
ex. ak= 1/(kx)
S100=
ex. ak= 2
S30=
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Writing the sum....figuring out the sequence
14 + 24 + 34 + 44 + 54 + 64 + 74 ex.
ex. √1 + √2 + √3 + √4 + √5 + ... + √76
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arithmetic series
Ch. 12 level 2
Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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The sum of an arithmetic sequence equals...!
Sn= a + (a+d) + (a+2d) +...+ [a+(n1)d]
= n 2(a + an)
2a + ann )(or
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If the sequence is:
Find the sum of the first 40 terms
3, 7, 11, 15...
Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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If the sequence is all odd numbers and an=81, find the series.
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Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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example. Triangle problems
level Total
How many squares will make up the triangle if there are 50 levels?
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geometric series
Ch. 12 level 2
Level 2 Arithmetic and Geometric Partial and Infinite Sums.notebook
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The formula for the sum of a geometric sequence
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Example: given the nth term and geometric partial sum
find the 4th partial sum, S4,
if a= 7, r = 2/3, and n=4
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Infinite SeriesTheoretically: its like eating cake, if you always only eat half of what is left, there is always some left. The total you eat never exceeds the whole cake or 1....
1/2 + 1/4 + 1/8 + 1/16 +...+ (1/2)n
Ʃ (1/2)n = 1 (1/2)k
Sn => 1 as n=> ∞
because as 2k gets really big, 1/2k will have a big denominator, making a smaller and smaller fraction off to subtracting 0
the for
mula!
IF Sn GETS CLOSE TO A FINITE NUMBER S, AS n GETS LARGE, THEN S IS THE SUM OF THE INFINITE SERIES:
IN THIS CASE, S = 1
1=Ʃ (1/2)n
n=1
∞
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Sum of an Inifinite Geometric Series
if |r|<1 (r is a fraction less than 1
a + ar+ ar2 + ar3 + ...+ arn1 + ...
Ʃ arn = a1 rn=1
∞
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Examples
ex. if a = 2 r = 1/5, what is the infinite sum?
its a straight line!
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Evaluation:
Practice! pg. Level 2 worksheet
1.
2.
if the sequence is 2,8,14,20... what is S20?
What is S100 of the geometric sequence:
a= 1, r = 1/2
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