Welcome Activity Write two different number pattern sequences that begin with the same two numbers.

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Presentation transcript:

Welcome Activity Write two different number pattern sequences that begin with the same two numbers.

HW Key: Practice Test 1 Diagnostic Level 1 B C D E

Arithmetic Sequences: Explicit Formula Unit 1 Chapter 11 Arithmetic Sequences: Explicit Formula

Objectives & HW: Students will be able define and use proper sequence notation, and write arithmetic sequences in explicit form. HW: p. 779: 2, 4, 8, 14, 16, 18, 20, 42

Definition of Terms Sequence – a function that computes an ordered list and has a set of natural numbers as its domain. Term – each number in a sequence

Definition of Terms Arithmetic Sequence – (or arithmetic progression) is a sequence in which the difference between each term and the preceding term is always constant. The common difference is denoted by d.

Sequence Notation a1 – the first term in a sequence a2 – the second term in a sequence an – the nth term, or the term in the nth position, of a sequence. an-1 – the term before an .

An arithmetic sequence is nothing more than a linear function with the specific domain of the natural numbers. The slope of the linear function is the common difference. The output of the function create the terms of the sequence.

Ex. 1: If sequence is arithmetic, find the common difference : 2, 8, 14, 20, . . . 72, 36, 18, . . . 12, 7, 2, -3, -8, . . . 6 Not arith. -5

Explicit Form of a Sequence Ex 2: Define the sequence -5, -1, 3, 7, . . . explicitly. The first term is a1 = -5. The common difference d = 4. a2 = a1 + 1d = -5 + 1(4) = -1 a3 = a1 + 2d = -5 + 2(4) = 3 a4 = a1 + 3d = -5 + 3(4) = 7, and so on. Hence, an = a1 + (n - 1)d an = -5 + (n - 1)4 an = -5 + 4n -4 an = -9 + 4n

Explicit Formula for the nth term of a Sequence: In an arithmetic sequence {an} an = a1 + (n – 1)d where d = common difference.

Students Try! Ex 3: Define the sequence -7, -4, -1, 2, . . . explicitly. a1 = -7, d = 3 an = -7 + (n – 1)(3) an = -7 + 3n – 3 an = -10 + 3n

Students Try! Ex 4: Find the 25th term of the given sequence: an = 13 – 2n a25 = 13 – 2(25) = 13 – 50 = -37

Finding Explicit Formula Ex 5: In an arithmetic sequence, a 5 = 8 and a12 = 29. Find d, a 1 , and an explicit formula for an. To find d:

(cont.) Ex 5: In an arithmetic sequence, a 5 = 8 and a12 = 29. Find d, a1 and an explicit formula for an. To find a1 : a5 = a1 + (n – 1)(d) 8 = a1 + (5 – 1)(3) 8 = a1 + 12 a1 = -4

To find an explicit formula: an = -4 + (n – 1)(3) an = -4 + 3n -3 (cont.) Ex 5: In an arithmetic sequence, a 5 = 8 and a12 = 29. Find d, a 1 and an explicit formula for an. To find an explicit formula: an = -4 + (n – 1)(3) an = -4 + 3n -3 an = -7 + 3n

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