1. Section 12-3 -Finding sums of arithmetic series -Using Sigma notation Taylor Morgan

2. Vocab. Arithmetic Sequence-(From previous section) sequence in which each term after the first is found by adding a constant, known as the common difference, d, to the previous term Series- an indicated sum of the terms in a sequence Arithmetic series- an indicated sum of the terms in an arithmetic sequence Ex. 1) 18, 22, 26, 30 Ex.2) 18+22+26+30 Ex. 3) -9,-3,3 Ex.4) 3/8+ 8/8 + 13/8 Arithmetic SequenceArithmetic Series

3. Terms in a Series represents the sum of the first n terms in a series So, for example, is the sum of the first four terms Ex. For the series 5+8+11+14+17, is 5+8+11+14 or 38.

4. To develop a formula for the sum of any arithmetic series, we will use the series below. Then, we will write in two different orders, one as the way above and the other in the opposite order, and add the equations.

5. Getting the Formula S =9/2(64) 9 Note that the sum had 9 terms The first and last terms of the sum are 64 2S =9(64) 9 S = 4+11+18+25+32+39+46+53+60 (+) S = 60+53+46+39+32+25+18+11+4 9 9 2S = 64+64+64+64+64+64+64+64+64 9

6. Where: d= the common difference by subtracting two consecutive terms n= number of terms a = 1 st term in a series a = nth term in a series 1 n Sum of an Arithmetic Series The sum of the first n terms of an arithmetic series is given by An arithmetic sequence S has n terms, and the sum of the first and last terms is a +a. Thus, the formula S =n/2(a +a ) represents the sum of any arithmetic series. 1n1nn n

7. Find the Sum of an Arithmetic Series Use either of the two sum formulas. Simplify. Multiply. Find the sum of the first 100 positive integers. The series is 1+2+3+…+100. Since you can see that a =1, a =100, and d=1, either sum formula can be used for this series. 1100 S =n/2(a +a ) S =100/2(1+100) S =50(101) S =5050 100 1nn Method 1Method 2 S =n/2[2a +(n-1)d] S =100/2[2(1) + (100-1)1] S =50(101) S =5050 100 n1

8. Find S for the arithmetic series with a =34, a =2, and n=9. Sum formula. Simplify. n 1n S =n/2(a + a ) nn 1 S =9/2(34+2) 9 S =162 9

9. Sigma Notation Since writing out series can be time- consuming and lengthy, a notation was devised to make writing them out easier, known as Sigma Notation. Ex. 3+6+9+12+…+30 can be expressed as: Σ 10 n=1 3n (This is read as the sum of 3n as n goes from 1 to 10) Last value of n First value of n Formula for the terms of the series

10. The variable, in this case n, is called the index of summation. To generate the terms of the series in sigma notation, replace the index of summation with the consecutive integers between the first and last values of the index. Ex. In this series, the values of n are 1, 2, 3, and so on, through 10. Σ 10 n=1 3n

11. Evaluating a Sum in Sigma Notation Evaluate: Start by finding the terms by replacing j with 5, 6, 7, and 8. Then add. Σ 8 (3j-4) J=5 Method 1: (The method described on the previous slide) Σ 8 (3j-4) J=5 = [3(5)-4] + [3(6)-4] + [3(7)-4] + [3(8) -4] =11+14+17+20 =62 So, the sum of the series is 62.

12. Evaluating a Sum in Sigma Notation (cont.) Σ 8 (3j-4) J=5 So, the sum of the series is 62. Method 2: Solving the same sum in Sigma notation Since the sum is an arithmetic series, use the formula S =n/2(a +a ). There are 4 terms, a =3(5)-4 or 11, and a =3(8)-4 or 20. S =4/2(11+20) S =62 n 1n 1 4 4 4

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