CK-12-Pre-Calculus Concepts

12.5. Geometric Series http://www.ck12.org 0 + 0 + 0 + 0 +··· Example C What is the sum of the first 8 terms in the following ge ...

http://www.ck12.org Chapter 12. Discrete Math a 1 = 100 ,r= 1. 06 The sum of the 10 years of deposits is: a 1 (^11 −−rrn)= 100 ( ...

12.6. Counting with Permutations and Combinations http://www.ck12.org 12.6 Counting with Permutations and Combina- tions Here yo ...

http://www.ck12.org Chapter 12. Discrete Math You can see where the six comes from by making a decision chart and using the Fund ...

12.6. Counting with Permutations and Combinations http://www.ck12.org are four options for the first passenger seat. Once that p ...

http://www.ck12.org Chapter 12. Discrete Math Adecision chartis a sequence of numbers that multiply together where each number r ...

12.6. Counting with Permutations and Combinations http://www.ck12.org You visit 12 colleges and want to apply to 4 of them. In ...

http://www.ck12.org Chapter 12. Discrete Math 12.7 Basic Probability Here you will calculate the probability of simple and compo ...

12.7. Basic Probability http://www.ck12.org P(EC) = 1 −P(E) You will often be looking at probabilities of two or more independen ...

http://www.ck12.org Chapter 12. Discrete Math 99100 ≈ 0. 366 Therefore, the probability that a plane does not crash in the next ...

12.7. Basic Probability http://www.ck12.org 20 · 0. 30 · 0. 1 = 0. 006 = 0 .6% The naïve approach would be to simply add the ...

http://www.ck12.org Chapter 12. Discrete Math 12.8 Binomial Theorem Here you will apply the Binomial Theorem to expand binomials ...

12.8. Binomial Theorem http://www.ck12.org Be extremely careful when working with binomials of the form(a−b)n. You need to remem ...

http://www.ck12.org Chapter 12. Discrete Math Vocabulary TheBinomial Theoremis a theorem that states how to expand binomials tha ...

12.8. Binomial Theorem http://www.ck12.org Collapse the following polynomials using the Binomial Theorem. 243x^5 − 405 x^4 + 27 ...

http://www.ck12.org Chapter 12. Discrete Math 12.9 Induction Proofs Here you will learn how to prove statements about numbers us ...

12.9. Induction Proofs http://www.ck12.org Inductive Hypothesis (starting equation): 2 + 22 +··· 2 k= 2 k+^1 − 2 Multiply by 2: ...

http://www.ck12.org Chapter 12. Discrete Math Base Case(s):Two base cases are shown however only one is actually necessary. 13 = ...

12.9. Induction Proofs http://www.ck12.org Vocabulary Thebase caseis the anchor step. It is the first domino to fall, creating a ...

http://www.ck12.org Chapter 12. Discrete Math Rewrite the right side: 1+ 2 + 3 + 4 +···+k+(k+ 1 ) =(k+^1 )(( 2 k+^1 )+^1 ) ∴ Pra ...