UNIT 12 SUM AND UNIT 12 SUM AND SIGMA NOTATIONSIGMA NOTATION

Warm UpFind the first 5 terms of each sequence.

1. 4.

2.

5.

3.

Warm Up ContinuedWrite a possible explicit rule for the nth term of each sequence.

6. 1, 2, 4, 8, 16, …

7. 4, 7, 10, 13, 16, …

Evaluate the sum of a series expressed in sigma notation.

Objective

seriespartial sumsummation notation

Vocabulary

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 asequence. 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.

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.

Check It Out! Example 2a

Expand each series and evaluate.

= (2(1) – 1) + (2(2) – 1) + (2(3) – 1) + (2(4) – 1)

= 1 + 3 + 5 + 7

= 16

Expand the series by replacing k.

Simplify.

Check It Out! Example 2b

Expand each series and evaluate.

= – 5 – 10 – 20 – 40 – 80

= –155

= –5(2)(1 – 1) – 5(2)(2 – 1) – 5(2)(3 – 1) – 5(2)(4 – 1) – 5(2)(5 – 1)

Expand the series by replacing k.

Simplify.

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.

Similar methods will help you find the sum of a quadratic series.

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

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

Check It Out! Example 3a

Evaluate the series.

Method 1 Use the summation formula.

Constant series

There are 60 terms.

= nc = 60(4) = 240

Method 2 Expand and evaluate.

= 60 + 60 + 60 + 60

4 items

= 240

Linear series

Check It Out! Example 3b

Evaluate each series.

Method 1 Use the summation formula.

= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120

Method 2 Expand and evaluate.

Check It Out! Example 3c

Evaluate the series.

= 385

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=

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?

11 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.

22 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.

Solve33

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 Back44

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

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