Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a...
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Transcript of Consider your summer job…..did you ever get a raise? Suppose you get paid $100.00 per week, with a...
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
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
Watch this……
+S = 100 + 105 + 110 + 115
S = 115 + 110 + 105 + 100 2S = 215 + 215 + 215 + 215 2S = 215(4)2S = 860
S = 430
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?
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
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
Sn = (mean of first and last term)
( n )2
(number of terms)
X
Sn = (a + tn) X
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 …
Sn = n [2a + (n – 1)d]2
n = term number
a = first term
d = common difference
Determine the sum:
1) 2 + 4 + 6 + 8 + 10 =30
Sn = n [2a + (n – 1)d]2
n = 5
a = 2
d = 2
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
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
tn = -94a = -4d = -6
Since tn = a + (n – 1)d,
-94 = -4 + (n – 1)(-6)-94 = -4 – 6n + 6-96 = - 6nn = 16
Find the sum: -4, -10, -16, …-94
Sn = n [2a + (n – 1)d]2
n = 16
a = -4
d = -6
Go back ….
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
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 + d a + 2d a + 3d
a a + d a + 2d a + 3d a + 4d
Generate a linear system with 2 equations and 2 unknowns
____ + ____ + ____ + ____ = -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
-
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 469
[1-3] a,c6,11,13,
15,19, 23quiz wedtest friday
Page 56
[1,3,5]a,c6a, 9a,c,e10,13,1719,20,22*