Warm-up 8-23-12 A field is 91.4 m long x 68.5 m wide. Calculate the area of the field in metres Calculate the area of the field in centimeters Express your answer to (b) in the form a x 10k where 1≤a<10 and 𝑘∈𝒁 6,260.9 m2 62,609,000 cm2 6.2609 x 107 cm2

The Great Leonardo Fibonacci This man was known best for introducing the “Hindu-Arabic” number system into Europe, and for his book “Liber Abbaci” which describes the rules we use in elementary school for adding, subtracting, multiplying, and dividing numbers

Arithmetic Sequences 2.5 - Arithmetic sequences and series, and their applications. Included: simple interest as an application. Link with simple interest 8.2. Use of the formulae for the nth term and the sum of the first n terms.

Number Sequences For each of the number sequences 4, 2, 0, -2, …. Describe the pattern Write down the next two terms in the sequences 4, 2, 0, -2, …. 1, 4, 9, 16, 25, … 100, 50, 25, 12.5, …

Answers to previous slide: i. each term is 2 less than the previous term ii. -4 & -6 i. Sequence of square numbers starting from 1 ii. 36 & 49 i. multiplying each previous term by 0.5 ii. 6.25 & 3.125

Number Sequences (cont.) Find the next two terms of these number sequences: 1, 8, 27, 64, … 0.4, 0.2, 0, -0.2, … 0, 1, 1, 2, 3, 5, …

Answers 125 & 216 This is the sequence of cube numbers starting with 1 -0.4 & -0.6 Each term is 0.2 less than the previous term 8 and 13 We obtain each term by adding the preceding two terms. This rule is valid as from the 3rd term on. This sequence is known as the Fibonacci sequence

General Form of a Number Seq. The terms of a sequence are designated in the following way: u1, u2, u3, u4,…,un, … This means that u1 represents the 1st term, u2 represents the 2nd term, etc. The nth term of the sequence is represented by un and is called the general term.

Writing a general term When writing a general term, you will have to look for a pattern and make sure the pattern will generate the sequence by plugging in the integers {1,2,3,4,5,…}. Write the nth term of the sequence: The sequence: {1, 4, 9, 16…} u1 = 1 u2 = 4 u3 = 9 u4 = 16 … un = n2

Writing a general term (cont.) The general term is a formula which is expressed in terms of n, a positive integer. This formula allows us to work out any term of the sequence. We make sure our general term (formula) works by plugging in the integers (1, 2, 3, etc…) and checking to see if this is giving us the numbers in our sequence (u1, u2, u3, u4,…,un, …)

Arithmetic Sequences These are special types of number sequences: A number sequence is arithmetic if there is a constant difference between each term and the previous one. This constant is called the common difference (d).

Examples Find the common difference for this arithmetic sequence: {5, 9, 13, 17 ... } Solution: d = 4 Find the common difference for the arithmetic sequence whose formula is: an = 6n + 3 Solution: {9, 15, 21, 27, 33, ...} The list shows the common difference to be 6. http://www.regentsprep.org/Regents/math/algtrig/ATP2/ArithSeq.htm

Formula for Writing General Form un = u1 + (n – 1)d Where n = place number in the sequence, and d = common difference This formula can be used to create the general form of an arithmetic sequence

Writing General Formula (practice) Write the general formula of the following sequences: {3, 7, 11, 15, …} 𝑈 𝑛 =3+ 𝑛−1 4 𝑈 𝑛 =4𝑛−1 {9, 4, -1, -6, …} 𝑈 𝑛 =9+ 𝑛−1 −5 𝑈 𝑛 =−5𝑛+14

Summing up a Number of Terms in an Arithmetic Sequence Find the sum of the first 25 terms of the sequence: 𝑈 𝑛 =6𝑛+2 𝑛=1 25 6𝑛+2 =2000 Find the sum of the first 106 terms of the sequence: {1,6,11,16,21,26,…} General term is: 𝑈 𝑛 =5𝑛−4 𝑛=1 106 5𝑛−4 =27,931

Homework Pg. 91-92 in the RED BOOK (#’s 1-12 all)

What is the Fibonacci Sequence http://en.wikipedia.org/wiki/Fibonacci_number Videos for the Fibonacci Sequence: http://www.youtube.com/watch?v=2pbEarwdusc http://www.youtube.com/watch?v=e9MwNm0gXd8