9.4 Arithmetic Series Notes

Pre­AP Algebra 2 9.4 Arithmetic Series *Objective: Define arithmetic series and find their sums

When you know two terms and the number of terms in a finite arithmetic sequence, you can find the sum of the terms. A series is the indicated sum of the terms of a sequence. A finite series has a first terms and a last term. An infinite series continues without end.

Finite Sequence Finite Series 6, 9, 12, 15, 18 6 + 9 + 12 + 15 + 18 = 60

Infinite Sequence Infinite Series 3, 7, 11, 15, ... 3 + 7 + 11 + 15 + ...

An arithmetic series is a series whose terms form an arithmetic sequence. When a series has a finite number of terms, you can use a formula involving the first and last term to evaluate the sum.

The sum Sn of a finite arithmetic series a1 + a2 + a3 + ... + an is n Sn = /2 (a1 + an) a1 : is the first term an : is the last term (nth term) n : is the number of terms in the series

Finding the Sum of a finite arithmetic series Ex1) a. What is the sum of the even integers from 2 to 100

b. what is the sum of the finite arithmetic series: 4 + 9 + 14 + 19 + 24 + ... + 99

c. What is the sum of the finite arithmetic series: 14 + 17 + 20 + 23 + ... + 116?

Using the sum of a finite arithmetic series Ex2) A company pays $10,000 bonus to salespeople at the end of their first 50 weeks if they make 10 sales in their first week, and then improve their sales numbers by two each week thereafter. One salesperson qualified for the bonus with the minimum possible number of sales. How many sales did the salesperson make in week 50? In all 50 weeks?

You can use the Greek capital letter sigma, Ʃ, to indicate sum. You use limits to indicate how many terms you are adding. Limits are the least to greatest values of n (number of terms) in the series. You write the limits below and above the to indicate the first and last number of terms of the series.

To find the number of terms in a series written in Ʃ form, subtract the lower limit from the upper limit and add 1. Following the sigma notation is the explicit formula of the sequence.

Writing a series in summation notation Ex3) a. 7 + 11 + 15 + ... + 203 + 207 b. ­5 + 2 + 9 + 16 + ... + 268

If the explicit formula for the nth term in summation notation is a LINEAR function of n, then the series is arithmetic. The slope of the linear function is the common difference between terms of the series.

Finding the Sum of a series

Ex4) a. Ʃ (5n + 3) b. Ʃ (­4n ­ 3)

c. Ʃ (3n ­ 8) d. Ʃ (­2n + 1)

On a graphing calculator, you can find the sum of a finite series by using Alpha Window (F2). Select 2: Ʃ