Download Alg2 Notes 9.5.notebook

Alg2 Notes 9.5.notebook
April 17, 2013
9­5 Infinite Geometric Series & Proof by Induction Skills we've learned, Quiz today 9.1­9.4
1. Write the rule for the sequence both recursively and explicitly if possible: 4, 8, 16, 32, 64, ...
2. Write in summation notation: 5 ­ 7 + 9 ­ 11 + 13
3. Evaluate
4. Find the 8th term of an geometric sequence if a3 = 9 and a5 = 81
Warm­up Answers
1. Write the rule for the sequence both recursively and explicitly if possible: 4, 8, 16, 32, 64, ...
2. Write in summation notation: 5 ­ 7 + 9 ­ 11 + 13
3. Evaluate
4. Find the 8th term of an geometric sequence if a3 = 9 and a5 = 81
Alg2 Notes 9.5.notebook
April 17, 2013
Got this figured out...
Company A is growing linearly. In 2010, Company A had 4000 employees, In 2011, Company A had 4200 employees, (+200)
In 2012, Company A had 4400 employees. (+200)
What is the average growth? 200/year: normal mean
Company B is growing exponential.
In 2010, Company B had 4000 employees,
In 2011, Company B had 4200 employees, (5%)
In 2012, Company B had 4620 employees (10%)
What is the average growth? 7.47% per year: geometric mean
Also, if Company B grew by 5%, then 10%, then 15%, you would do the cube root instead of the square root! 9.92% per year
Another example: What is the mean dimensions for a cube to have the same volume as a rectangular prism of sides 5, 10, and 15?
Another example: Altitude (x) is the geometric mean of a and b
x
a
b
Wikipedia: The geometric mean applies only to positive numbers. 9­5 Infinite Geometric Series & Proof by Induction Find sums of infinite geometric series.
Use mathematical induction to prove statements.
Calculus (areas), Proofs
Alg2 Notes 9.5.notebook
April 17, 2013
I. Infinite Series: Convergent or Divergent
Take the series and the series Look at a graph of the partial sums for both of these series:
When , the series converges.
When , the series diverges.
1. Determine whether each geometric series converges or diverges.
Alg2 Notes 9.5.notebook
April 17, 2013
II. The Sum of an Infinite Geometric Series 2. Find the sum of the infinite geometric series
(if it exists)
3. Find the sum of the infinite geometric series
(if it exists)
A.
B.
Alg2 Notes 9.5.notebook
April 17, 2013
III. Repeating Decimals as Fractions
4. Write 0.63 as a fraction in simplest form.
Book way:
Step 1 Write the repeating decimal as an infinite geometric series.
Step 2 Find the common ratio.
r = .01
Step 3 Find the sum.
My way:
5. Write as a fraction in simplest form.
A. 0.123123123...
B. 0.999999....
Alg2 Notes 9.5.notebook
April 17, 2013
IV. Proofs by Mathematical Induction IV. Proofs by Mathematical Induction 6. Use mathematical induction to prove that 1 + 4 + 7 + .... + (3n 2) = –
Step 1 Base case: Show that the statement is true for n = 1.
Step 2 Assume that the statement is true for a natural number k.
Step 3 Prove that it is true for the natural number k + 1.
Therefore, 1 + 4 + 7 + .... + (3n 2) = –
Alg2 Notes 9.5.notebook
April 17, 2013
7. Use mathematical induction to prove that the sum of the first n odd numbers is 1 + 3 + 5 + … +(2n ­ 1) = n2.
Step 1 Base case: Show that the statement is true for n = 1.
2(1) – 1 = 12
Step 2 Assume that the statement is true for a natural number k.
Step 3 Prove that it is true for the natural number k + 1.
V. Counterexamples 8. Identify a counterexample to disprove a3 > a2, where a is a whole number.
9. Identify a counterexample to disprove
where a is a real number.
Alg2 Notes 9.5.notebook
April 17, 2013
9.5 p.668 #1­ 14, 15 ­ 39 (x3), 46, 54, 58, 61, 63, 64, 68