Warm-Up Honors Algebra 2 9/7/18

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Warm-Up Honors Algebra 2 9/7/18 Decide whether each sequence is arithmetic or geometric. State the common difference or common ratio. 4, 10, 16, 22, 28, … 5, 8, 11, 14, … 4, 12, 36, 108, … 2, -4, 8, -16, …

INTRO TO SEQUENCES AND SERIES What is a “term”?

INTRO TO SEQUENCES AND SERIES What is a “term”? A specific number in a sequence or series. a1= first term a2= second term an=nth term (or last term)

ex1. The nth term of a sequence is given by: an = n2 + 2 a) Write out the first 5 terms.

ex 1 (continued) The nth term of a sequence is given by: an = n2 + 2 b) What is the value of the 7th term?

ex1. (continued) The nth term of a sequence is given by: an = n2 + 2 c) Find a9.

ex2. The nth term of a sequence is given by: an = 4(n + 2)(n – 1) Use the table function of the graphing utility on your calculator to write out the first 5 terms.

INTRO TO SEQUENCES AND SERIES What is a “Recursively defined Sequence”? A sequence in which calculating each term is based on the value of the term before.

INTRO TO SEQUENCES AND SERIES Recursively defined Sequence Find the first six terms of the “famous” sequence described below

Important Formulas for an Arithmetic Sequence: Recursive Formula Explicit Formula 𝑎 1 = ? an = (an – 1 ) +d 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 Where: an is the nth term in the sequence a1 is the first term n is the number of the term d is the common difference

Important Formulas for a Geometric Sequence: Recursive Formula Explicit Formula 𝑎 1 = ? an = (an – 1 ) r 𝑎 𝑛 = 𝑎 1 ∙ 𝑟 𝑛−1 Where: an is the nth term in the sequence a1 is the first term n is the number of the term r is the common ratio

In the sequence 10, 40, 70, 100, …. The constant difference between the terms is 30 The first term of the sequence is 10 The recursive formula for this sequence would be: **I substituted in 10 for the first term and 30 for the constant difference**

Write the recursive formula of the sequence 4, 7, 10, 13, ….

In the sequence 4, 7, 10, 13, …. To find the 5th term recursively, I substitute it into the formula I just made: an = an-1 + 3 a5 = a5-1 + 3 in words: 5th term equals the 4th term plus 3 a5 = 13 + 3 a5 = 16

The constant difference between the terms is 30 In the sequence 10, 40, 70, 100, …. The constant difference between the terms is 30 The first term of the sequence is 10 The explicit formula for this sequence would be: an = 10 + 30( n - 1) which simplifies to: an = -20 + 30n **I substituted in 10 for the first term and 30 for the constant difference and distributed**

Write the explicit formula of the sequence 4, 7, 10, 13, ….

In the sequence 4, 7, 10, 13, …. To find the 11th term explicitly, I substitute in the nth term into the formula I just made: an = 1 + 3n a11 = 1 + 3(11) a11 = 34

Find the 15th term of the sequence using the formula: an = 1 + 3n

Geometric Sequence Ex: Write the explicit formula for the sequence 9, 3, 1, …. Write the recursive formula for the sequence 9, 3, 1, ….