Chapter 11 Sequences and Series

11.1 Arithmetic Sequences

Key Terms Sequence: an ordered list of numbers 1, 3, 5, 7, 9, …. Terms: the numbers in the sequence The 1st term in the sequence above is 1, 2nd is 3, 3rd is 5, etc Arithmetic Sequence: a sequence in which each terms after the 1st is found by adding a constant Common difference: the number added to each term in an arithmetic sequence

Arithmetic sequence Formula: an = a1 + (n – 1)d an = specific term (nth term) a1 = 1st term in the sequence n = number of position for given term d = common difference

Find the next four terms in the sequence. -6, -2, 2,…. 1.6, 1.1, 0.6, ….

Write an equation for the nth term 10, 7, 4, 1, …

Write an equation for the nth term 8, 17, 26, 35, …

The table shows typical costs for a construction company to rent a crane for one, two, three, and four months. If the sequence continues, how much would it cost to rent the crane for a year? Months Cost ($) 1 75,000 2 90,000 3 105,000 4 120,000 The construction company has a budget of $350,000 for crane rental. The job is expected to last 18 months. Will the company be able to afford the crane rental for the entire job? Explain.

Find the first 5 terms for the sequence described Find the indicated term for the sequence a1 = 2 d = 13 a1 = 20, d = 4, n = 81

Arithmetic Means The terms between any two non-successive terms in an arithmetic sequence Use the formula to find the common difference Then use the common difference to calculate the arithmetic means

Find the arithmetic means between the given terms 16,__, __, __, __, 91, … 15.6, __, __, __, 60.4, ...

11.2 Arithmetic Series

What is a series? When the terms of a sequence are added, the indicated sum of the terms is called a series. Example Sequence 1, 2, 3, 4, 5, 6 Series 1+2+3+4+5+6

Sum of an Arithmetic Series  

Find the sum of the series 7+12+17+22+…+52 n = 10

Find the sum of the first 50 positive even integers

Aiden did pushups every day in January Aiden did pushups every day in January. He started on January 1st and increased the number of pushups done each day by 1. He did a total of 1085 pushups for the month. How many pushups did he do on the first day?

Find the first three terms a1 = 9 an = 105 Sn = 741 Step 1: plug in to find n Step 2: plug in to find d Step 3: use d to find the terms

Sigma Notation Compact form for series Uses the summation symbol Σ Greek letter sigma

Sigma Notation formula for the sequence Last value of n First value of n

Ex: For the sequence 1, 2, 3, 4, 5, 6 a1 = 1 d = 1 an = 1 + (n – 1)1  

Find the sum 8 5-2n n=1

Find the sum 150 11 + 2k k=1

11.3 Geometric Sequences

Geometric Sequence  

Find the next two terms in the sequence 405, 135, 45,…. 16, 24, 36, ….

Geometric Sequence Formula: an=a1rn-1  

Find a term given the first term and the common ratio a1 = -3 r = -2 Find the eighth term  

Write an equation for the nth term 432, 72,12, ….

Write an equation for the nth term 72, 56, 40, 24, …

Find a term given one term and the common ratio a4 = 108 r = 3 Find the tenth term. aA3 = 16 r = 4 Find the eighth term

Geometric Means The terms between any two non-successive terms in a geometric sequence Use the formula to find the common ratio Then use the common ratio to calculate the geometric means

Find the geometric means 2.25, __, __, __, 576 Find the two geometric means between 4 and 13.5

11.4 Geometric Series

Sum of a Geometric Series  

Find the sum for the first 10 terms a1 = 5, r = 4

Maria arranges some rows of dominoes so that after she knocks over the first one, each domino knocks over two more dominoes. If there are ten rows, how many dominoes does Maria use?

Evaluate  

Find the first term in a series S8 = 39,360 and r = 3

Find the sum of a geometric series for which