Essential Questions Series and Summation Notation How do we find the terms of an arithmetic sequence? How do we find the sum of an arithmetic series? Holt McDougal Algebra 2 Holt Algebra 2

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.

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

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

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

Using Summation Formulas Evaluate the series. 3. Quadratic series Method 1 Use the summation formula. Method 2 Use a calculator. n(n + 1)(2n + 1) 6 = 12(12 + 1)(2 · 12 + 1) 6 = (156)(25) 6 = = 650

Using Summation Formulas Evaluate the series. 4. Constant series Method 2 Expand and evaluate. Method 1 Use the summation formula. = 4 + 4 + 4 + 4 + 4 + 4 + 4 + . . . + 4 60 items There are 60 terms. = nc = 60(4) = 240 = 240

Using Summation Formulas Evaluate the series. 5. 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

Using Summation Formulas Evaluate the series. 6. Quadratic series Method 1 Use the summation formula. Method 2 Use a calculator. n(n + 1)(2n + 1) 6 = 10(10 + 1)(2 · 10 + 1) 6 = (110)(21) 6 = = 385

Lesson 5.2 Practice B