Arithmetic Series

Definition of an arithmetic series. The sum of the terms in an arithmetic sequence.

Arithmetic Sequence Arithmetic Series 4, 7, 10, , -4, ,,,

Example 1. During Kwanzaa one candle is lit the first night, two on the second night, and so forth for seven nights. How many candles are lit in all = 28 The symbol S n is used to represent the sum of the first n terms of a series. The above represents S 7.

Suppose we write S 7 in two different orders and find the sum S 7 = S 7 = S 7 = sums of 8

S 7 = S 7 = S 7 = sums of 8 2S 7 = 7(8) S 7 = 7(8) 2 Now analyze this expression in terms of S n.

S 7 = 7(8) 2 Now analyze this expression in terms of S n. 7 represents n and 8 represents the sum of the first and last terms, a 1 +a n. Thus we can replace the expression with S n = n(a 1 + a n ) 2

This formula can be used to find the sum of any arithmetic series. Sum of an arithmetic series. The sum of the first n terms of an arithmetic series is given by S n = n(a 1 + a n ) 2 where n is a positive integer.

Example 1. Find the sum of the first 50 positive even integers. = 25(102) S n = n(a 1 + a n ) 2 S 50 = 50( ) 2 = 2550

We have discovered in a previous lesson that a n = a 1 + (n-1)d Using this formula and substitution gives us another version of the formula for the sum of an arithmetic sequence.

S n = n(a 1 + a n ) 2 replace a n S n = n(a 1 + {a 1 +(n-1)d}) 2 S n = n(2a 1 +(n-1)d) 2 This formula is useful when we do not know the value of the last term.

Example 2. Find the sum of the first 40 terms of an arithmetic series in which a 1 = 70 and d = -21 The series is S n = n(2a 1 +(n-1)d) 2

Example 2. Sum of first 40 terms a 1 = 70 and d = -21 The series is S n = n(2a 1 +(n-1)d) 2 S n = 40(2(70) +(40-1)(-21)) 2 =

Example 3. Physics When an object is dropped it falls 16 feet in the first second, 48 feet in the second second, and 80 feet in the third second. How many feet would if fall in the 20th second.

Example 3. Object falling 16, 48, and 80 feet in 1, 2, and 3 seconds. How many feet would if fall in the 20th second Common difference is 32

Example 3. Object falling 16, 48, and 80 feet in 1, 2, and 3 seconds. How many feet would if fall in the 20th second. d = 32 a n = a 1 + (n-1)d a 20 = 16 + (20-1)32 a 20 = 624

Example 4. Refer to example 3 How many feet would a free falling object fall in 20 seconds? S n = n(2a 1 +(n-1)d) 2 S 20 = 20(2(16) + (20-1)(32)) 2 S 20 = 6400

Example 5. Find the first three terms of an arithmetic series in which a 1 = 13, a n = 157, and S n = 1445 We are given a 1, a n, and S n. Therefore we use the formula S n = n(a 1 + a n ) 2 and solve for n.

Example 5. Find the first 3 terms if a 1 = 13, a n = 157, and S n = 1445 S n = n(a 1 + a n ) 2 solve for n = n( ) = n(170) = 85n n = 17

Example 5. Find the first 3 terms if a 1 = 13, a n = 157, and S n = 1445 n = 17, now find d. a n = a 1 + (n-1)d 157 = 13 + (17-1)d 157 = d 144 = 16d d = 9

Example 5. Find the first 3 terms if a 1 = 13, a n = 157, and S n = 1445 n = 17, d = 9. a 2 = a 1 + d a 2 = a 2 = 22 a 3 = a 2 + d a 3 = a 3 = 31 The first 3 terms are 13, 22, and 31.

To simplify writing out series we use sigma or summation notation is written ∑ 2n. 10 n=1 This expression is read the sum of 2n as n increases from one to ten.

Last value of n First value of n ∑ 2n. 10 n=1 Formula for the seq The variable below the ∑ sigma is called the index of summation. The upper limit is the upper limit of the index.

∑ 2n. 10 n=1 The variable below the ∑ sigma is called the index of summation. The upper limit is the upper limit of the index. To generate the terms of the series, successively replace the index of summation with consecutive integers as values of n. In this series n = 1,2,3, and so on, through 10.

Example 6. Write the terms of and find the sum. ∑ (2k + 5) 7 k=3 ∑ (2k + 5) 7 k=3 = (23+5) + (24+5) + (25+5) + (26+5) + (27+5) = = 75