Series and Summation Notation 12-2 Series and Summation Notation Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Find the first 5 terms of each sequence. 1. 4. 2. 5. 3.

Warm Up Continued Write a possible explicit rule for the nth term of each sequence. 6. 1, 2, 4, 8, 16, … 7. 4, 7, 10, 13, 16, …

Objective Evaluate the sum of a series expressed in sigma notation.

Vocabulary series partial sum summation notation

In Lesson 12-1, you learned how to find the nth term of a sequence In Lesson 12-1, you learned how to find the nth term of a sequence. Often we are also interested in the sum of a certain number of terms of a sequence. A series is the indicated sum of the terms of a sequence. Some examples are shown in the table.

Because many sequences are infinite and do not have defined sums, we often find partial sums. A partial sum, indicated by Sn, is the sum of a specified number of terms of a sequence.

A series can also be represented by using summation notation, which uses the Greek letter  (capital sigma) to denote the sum of a sequence defined by a rule, as shown.

Example 1A: Using Summation Notation Write the series in summation notation. 4 + 8 + 12 + 16 + 20 Find a rule for the kth term of the sequence. ak = 4k Explicit formula Write the notation for the first 5 terms. Summation notation

Example 1B: Using Summation Notation Write the series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 6 terms. Summation notation.

Caution!

Check It Out! Example 1a Write each series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 5 terms. Summation notation.

Check It Out! Example 1b Write the series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 6 terms. Summation notation.

Example 2A: Evaluating a Series Expand the series and evaluate. Expand the series by replacing k. Evaluate powers. Simplify.

Example 2B: Evaluating a Series Expand the series and evaluate. = (12 – 10) + (22 – 10) + (32 – 10) + (42 – 10) + (52 – 10) + (62 – 10) Expand. = –9 – 6 – 1 + 6 + 15 + 26 Simplify. = 31

Check It Out! Example 2a Expand each series and evaluate. Expand the series by replacing k. = (2(1) – 1) + (2(2) – 1) + (2(3) – 1) + (2(4) – 1) = 1 + 3 + 5 + 7 Simplify. = 16

Check It Out! Example 2b Expand each series and evaluate. Expand the series by replacing k. = –5(2)(1 – 1) – 5(2)(2 – 1) – 5(2)(3 – 1) – 5(2)(4 – 1) – 5(2)(5 – 1) = – 5 – 10 – 20 – 40 – 80 Simplify. = –155

Finding the sum of a series with many terms can be tedious Finding the sum of a series with many terms can be tedious. You can derive formulas for the sums of some common series. In a constant series, such as 3 + 3 + 3 + 3 + 3, each term has the same value. The formula for the sum of a constant series is as shown.

The formula for the sum of a constant series is as shown.

A linear series is a counting series, such as the sum of the first 10 natural numbers. Examine when the terms are rearranged.

Notice that 5 is half of the number of terms and 11 represents the sum of the first and the last term, 1 + 10. This suggests that the sum of a linear series is , which can be written as Similar methods will help you find the sum of a quadratic series.

When counting the number of terms, you must include both the first and the last. For example, has six terms, not five. k = 5, 6, 7, 8, 9, 10 Caution

Example 3A: Using Summation Formulas Evaluate the series. Constant series Method 1 Use the summation formula. Method 2 Expand and evaluate. There are 7 terms.

Example 3B: Using Summation Formulas Evaluate the series. Linear series Method 1 Use the summation formula. Method 2 Expand and evaluate.

Example 3C: Using Summation Formulas Evaluate the series. Quadratic series Method 1 Use the summation formula. Method 2 Use a graphing calculator.

Method 2 Expand and evaluate. Method 1 Use the summation formula. Check It Out! Example 3a Evaluate the series. Constant series Method 2 Expand and evaluate. Method 1 Use the summation formula. There are 60 terms. = 60 + 60 + 60 + 60 4 items = nc = 60(4) = 240 = 240

Check It Out! Example 3b Evaluate each series. Linear series Method 1 Use the summation formula. Method 2 Expand and evaluate. = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120

Check It Out! Example 3c Evaluate the series. Quadratic series Method 1 Use the summation formula. Method 2 Use a graphing calculator. n(n + 1)(2n + 1) 6 = 10(10 + 1)(2 · 10 + 1) 6 = (110)(21) 6 = = 385

Example 4: Problem-Solving Application Sam is laying out patio stones in a triangular pattern. The first row has 2 stones and each row has 2 additional stones, as shown below. How many complete rows can he make with a box of 144 stones?

Understand the Problem 1 Understand the Problem The answer will be the number of complete rows. List the important information: • The first row has 2 stones. • Each row has 2 additional stones • He has 144 stones. • The patio should have as many complete rows as possible.

2 Make a Plan Make a diagram of the patio to better understand the problem. Find a pattern for the number of stones in each row. Write and evaluate the series.

Solve 3 Use the given diagram to represent the problem. The number of stones increases by 2 in each row. Write a series to represent the total number of stones in n rows.

Solve 3 Where k is the row number and n is the total number of rows. Evaluate the series for several n-values. 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) = = 110 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) + 2(11) = = 132

Solve 3 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) + 2(11) + 2(12) = 156 Because Sam has only 144 stones, the patio can have at most 11 complete rows.

Look Back 4 Use the diagram to continue the pattern. The 11th row would have 22 stones. S11 = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 = 132 The next row would have 24 stones, so the total would be more than 144.

Check It Out! Example 4 A flexible garden hose is coiled for storage. Each subsequent loop is 6 inches longer than the preceding loop, and the innermost loop is 34 inches long. If there are 6 loops, how long is the hose?

Understand the Problem 1 Understand the Problem The answer will be the total length of the hose. List the important information: • The first loop is 34 inches long. • Each subsequent loop is 6 inches longer than the previous one. • There are 6 loops.

2 Make a Plan Make a diagram of the hose to better understand the problem. Find a pattern for the length of each loop. Write and evaluate the series.

Solve 3 Use the given diagram to represent the problem. The first loop is 34 in. Each subsequent loop increases by 6 in. (34 + 6(1 – 1)) + (34 + 6(2 – 1)) + (34 + 6(3 – 1)) + (34 + 6(4 – 1)) + (34 + 6(5 – 1)) + (34 + 6(6 – 1)) = 294 in.

Look Back 4 Use the diagram to continue the pattern. The 6th loop would be 294 inches. S6 = 34 + 40 + 46 + 52 + 58 + 64 = 294.

Lesson Quiz: Part I Write each series in summation notation. 1. 1 – 10 + 100 – 1000 + 10,000 2. Write each series in summation notation. 55 3. 5. 325 64 4. 6. 285

Lesson Quiz: Part II 7. Ann is making a display of hand-held computer games. There will be 1 game on top. Each row will have 8 additional games. She wants the display to have as many rows as possible with 100 games. How many rows will Ann’s display have? 5