Consider your summer job…..did you ever get a raise? Suppose you get paid $ per week, with a $5 raise each week. How much will you have at the end of the summer (4 weeks)? 100, 105, 110, 115

To find a total, create an arithmetic series With small amounts, it is easy to calculate the totals manually. However, as we study more complex relationships, a more efficient method is needed… = 430

Watch this…… + S = S = S = S = 215(4) 2S = 860 S = 430

There is an even quicker way.. Find the mean of the first and last term ( ) = Multiply by the number of terms X 4 = 430 HOW?

Find the sum of 10,15,20,25, = =20 20 X 5 = Consider the numbers forming a bar graph…. 20

If we extend this approach to the general case, we can then generate a formula that will allow us to find the sum of a series of any length!

Sum of an Arithmetic Series S n = (mean of first and last term) ( n ) 2 (number of terms) X S n = (a + t n ) X

Since t n = a + (n – 1)d, - make the substitution ( n ) 2 S n = (a + t n ) X ( n ) 2 S n = [a + a + (n – 1)d] X Collect the a’s and multiply by n …

S n = n [2a + (n – 1)d] 2 n = term number a = first term d = common difference

Determine the sum: 1) =30 S n = n [2a + (n – 1)d] 2 n = 5 a = 2 d = 2

S n = n [2a + (n – 1)d] 2 n = 5, a = 2, d = 2 S 5 = 5 [2(2) + (5 – 1)2] 2 = 5 [4 + 8] 2 = 5 [12] 2 = 30

Find the sum: -4, -10, -16, …-94 S n = n [2a + (n – 1)d] 2 n = ? a = -4 d = -6 Use the term formula to find n

t n = -94 a = -4 d = -6 Since t n = a + (n – 1)d, -94 = -4 + (n – 1)(-6) -94 = -4 – 6n = - 6n n = 16

Find the sum: -4, -10, -16, …-94 S n = n [2a + (n – 1)d] 2 n = 16 a = -4 d = -6 Go back ….

S n = n [2a + (n – 1)d] 2 n = 16, a = -4, d = -6 S 16 = 16 [2(-4) + (16 – 1)(-6)] 2 = 16 [(-8) + (-90)] 2 = 8 [-98] = -784

The sum of the first 4 terms of an arithmetic series is –8 and the sum of the first 5 terms is 85. Determine the first term and the common difference.

____ + ____ + ____ + ____ = -8 Let the first term be “a”, let the common difference be “d” ____ + ____ + ____ + ____ + ____ = 85 a a + da + 2da + 3d aa + da + 2d a + 3da + 4d Generate a linear system with 2 equations and 2 unknowns

____ + ____ + ____ + ____ = -8 a + a + d + a + 2d + a + 3d = -8 a + a + d + a + 2d + a + 3d + a + 4d = 85 ____ + ____ + ____ + ____ + ____ = 85 a a + da + 2da + 3d aa + da + 2d a + 3da + 4d 4a + 6d = -8 5a + 10d = 85

- X 5 X 4 4a + 6d = -8 5a + 10d = 85 20a + 30d = a + 40d = d = -380 d = 38 4a + 6(38) = -8 4a = -8 a = -59 Substitute

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