Sequences and Series 13.3 The arithmetic sequence

DO NOW: Given the sequence 4, 15, 32, 55, 84, 119…

a.Is it linear or quadratic?

b.Write a recursive formula.

c.Write an explicit formula.

Arithmetic Sequences Example #1

n 1 2 3 4 5a 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. 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 n

n

a aS n

1

2

There’s MORE!

nn

a aS n

1

2If and a n = a 1 + (n-1)d ,

then Sn =

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 terms2,4,6,8,….2n

Arithmetic series – the sum of the list2 + 4 + 6 + 8…+ 2n +…

Finite series – also a partial sum2 + 4 + 6 + 8 = 20

Infinite series - 2 + 4 + 6 + 8…+ 2n +…