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Page 1: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the
Page 2: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Consider your summer job…..did you ever get a raise?

Suppose you get paid $100.00 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

Page 3: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

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…..

100 + 105 + 110 + 115 = 430

Page 4: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Watch this……

+S = 100 + 105 + 110 + 115

S = 115 + 110 + 105 + 100 2S = 215 + 215 + 215 + 215 2S = 215(4)2S = 860

S = 430

Page 5: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

There is an even quicker way..Find the mean of the first and

last term

(100 + 115) = 107.5 2

Multiply by the number of terms

107.5 X 4 = 430HOW?

Page 6: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Find the sum of 10,15,20,25,30

10 + 15 + 20 + 25 + 30 = 100

30 + 102

=2020 X 5

= 10020 + 20 + 20 + 20 + 20

Consider the numbers forming a bar graph….

20

Page 7: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

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!

Page 8: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Sum of an Arithmetic Series

Sn = (mean of first and last term)

( n )2

(number of terms)

X

Sn = (a + tn) X

Page 9: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Since tn = a + (n – 1)d,

- make the substitution

( n )2

Sn = (a + tn) X

( n )2

Sn = [a + a + (n – 1)d] X

Collect the a’s and multiply by n …

Page 10: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Sn = n [2a + (n – 1)d]2

n = term number

a = first term

d = common difference

Page 11: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Determine the sum:

1) 2 + 4 + 6 + 8 + 10 =30

Sn = n [2a + (n – 1)d]2

n = 5

a = 2

d = 2

Page 12: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Sn = n [2a + (n – 1)d]2

n = 5, a = 2, d = 2

S5 = 5 [2(2) + (5 – 1)2]2

= 5 [4 + 8]2

= 5 [12]2

= 30

Page 13: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Find the sum: -4, -10, -16, …-94

Sn = n [2a + (n – 1)d]2

n = ?

a = -4

d = -6 Use the term formula to find n

Page 14: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

tn = -94a = -4d = -6

Since tn = a + (n – 1)d,

-94 = -4 + (n – 1)(-6)-94 = -4 – 6n + 6-96 = - 6nn = 16

Page 15: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Find the sum: -4, -10, -16, …-94

Sn = n [2a + (n – 1)d]2

n = 16

a = -4

d = -6

Go back ….

Page 16: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Sn = n [2a + (n – 1)d]2

n = 16, a = -4, d = -6

S16 = 16 [2(-4) + (16 – 1)(-6)]2

= 16 [(-8) + (-90)]2

= 8 [-98] = -784

Page 17: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

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.

Page 18: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

____ + ____ + ____ + ____ = -8

Let the first term be “a”, let the common difference be “d”

____ + ____ + ____ + ____ + ____ = 85

a a + d a + 2d a + 3d

a a + d a + 2d a + 3d a + 4d

Generate a linear system with 2 equations and 2 unknowns

Page 19: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

____ + ____ + ____ + ____ = -8

a + a + d + a + 2d + a + 3d = -8a + a + d + a + 2d + a + 3d + a + 4d = 85

____ + ____ + ____ + ____ + ____ = 85

a a + d a + 2d a + 3d

a a + d a + 2d a + 3d a + 4d

4a + 6d = -85a + 10d = 85

Page 20: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

-

X 5

X 4

4a + 6d = -85a + 10d = 85

20a + 30d = -4020a + 40d = 340

- 10d = -380

d = 38

4a + 6(38) = -8 4a + 228 = -8 a = -59

Substitute

Page 21: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

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15,19, 23quiz wedtest friday

Page 22: Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a $5 raise each week. How much will you have at the

Page 56

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