Sequences and Series 13.3 The arithmetic sequence DO NOW: Given the sequence 4, 15, 32, 55, 84, 119… Is it linear or quadratic? Write a recursive formula. Write an explicit formula.

Arithmetic Sequences Example #1 a n or a(n) 5 9 13 17 21 a) Write a recursive definition. b) Write a closed-form (explicit) definition. c) Common difference is ___. a n = a 1 + (n-1)d

Example #2: The 10th term of an arithmetic sequence is 146 and the 18th term is 98. Find the first term and common difference.

Example #3: A portion of the arithmetic sequence is given Example #3: A portion of the arithmetic sequence is given. Fill in blanks. 28, ___, ____, ____, 42 This is also known as finding the arithmetic means. “Find 3 arithmetic means between 28 and 42.”

Arithmetic Series & Sigma Notation Example #4: Write the finite series in Sigma Notation Finite series – also a partial sum. S4 = 2 + 4 + 6 + 8 = 20

Example #5: Find the partial sum S10 of the first ten terms of the arithmetic sequence. an = 2 + (n-1)3 S10 = The sum of the first n terms, Sn , of the arithmetic sequence an , with common difference d is

There’s MORE! If and a n = a 1 + (n-1)d , then Sn =

Find the sum of the first 75 terms of the sequence. Example #6: An arithmetic sequence has a1 = -10 and a common difference of 0.25. Find the sum of the first 75 terms of the sequence.

Arithmetic Series Arithmetic sequence – the list of terms 2,4,6,8,….2n Arithmetic series – the sum of the list 2 + 4 + 6 + 8…+ 2n +… Finite series – also a partial sum 2 + 4 + 6 + 8 = 20 Infinite series - 2 + 4 + 6 + 8…+ 2n +…