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**Unit 6: Sequences & Series**

LG 6-1: Arithmetic & Geometric Sequences LG 6-2: Limits of Sequences LG 6-3: Partial & Infinite Series Test & Project Due 3/08

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**Arithmetic vs. Geometric**

Arithmetic sequences – the terms are found by adding or subtracting a common number to each subsequent term. Linear functions. Geometric sequences – the terms are found by multiplying a common ratio to each subsequent term. Exponential functions

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Arithmetic Sequence Each term is found by adding some constant real number. The constant real number is called the common difference.

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The nth Term of an A.S. d is the common difference a0 is the 0th term

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**Practice Determine if the sequence is arithmetic.**

If yes, find the common difference. Then, write the explicit formula.

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**Example: Find the nth Term**

Find a formula for the nth term of the A.S. whose common difference is 3 and whose first term is 2.

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Discussion Find the 100th term of this sequence:

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**Find the indicated term of each sequence.**

Practice Find the indicated term of each sequence.

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**Example: Find the Terms**

The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first several terms of this sequence.

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Recursive Formula If you know any term of an arithmetic sequence and you know the common difference of the sequence, you can find the next term.

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You try! Write a recursive and an explicit formula for the arithmetic sequence 17, 21, 25, … An arithmetic sequence has first term 7 and common difference 3. Write the first six terms of the series.

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Geometric Sequences Geometric Sequence– a sequence whose consecutive terms have a common ratio. A sequence is geometric if and only if the ratios of consecutive terms are the same. The number r is the common ratio.

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Geometric Sequences Explicit formula Recursive formula an = a1(r)n – 1

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**Find a formula for the nth term.**

5, 15, 45, … an = a1rn – 1 an = 5(3)n – 1 What is the 9th term? an = 5(3)n – 1 a9 = 5(3)8 a9 = 32805

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**Write the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3).**

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**Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05**

an = a1rn – 1 a15 = (20)(1.05)15 – 1 a15 =

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**LG 6-1 Sequences: Summary**

What is the difference between an arithmetic and a geometric sequence? How do you find the common difference? How do you find the common ratio? Write the nth term of this sequence: 5, 18, 31, 44, … Find the 26th term of this sequence: 42, 14, 4.67, …

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**Ticket out the Door (TOTD)**

GIRLS 36, 27, 18, 9, … BOYS 2, 16, 128, 1024, … Is your sequence arithmetic or geometric sequence? Does your sequence have a common difference or common ratio? Write the nth term of your sequence. Find the 15th term of your sequence.

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