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

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

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

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

Practice Determine if the sequence is arithmetic. If yes, find the common difference. Then, write the explicit formula.

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.

Discussion Find the 100th term of this sequence:

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

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.

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.

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.

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.

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

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

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

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 = 39.599

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, …

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.