Download Arithmetic Series

Warm up
7, 15, 23, 31…
 Is this sequence arithmetic or geometric?
Arithmetic
 What is the recursive definition?
f(1) = 7
f(n) = f(n-1) + 8
 What is the explicit formula?
f(n) = 7 + (n-1)8
9-4
Arithmetic Series
Today’s Objective:
I can find the sum of an arithmetic series.
The first four rows of chairs are set up
for a meeting. The seating pattern is to
continue through 20 rows. How many
chairs will there be in all 20 rows?
4 + 5 + 6 + 7 + . . . + 23 = 270
20
( 4 + 23 ) = 270
𝑆20 =
2
Series: Sum of the terms of a sequence
2 + 4 + 6 + ··· + 100
50
( 2 +100 )
𝑆50 =
2
Finite Series:
Infinite Series:
Has first and last term Continues without end
6 + 9 + 12 + 15 + 18 3 + 7 + 11 + 15 + . . .
𝑛
Sum of a Finite Arithmetic Series
𝑆𝑛 = (𝑓(1) + 𝑓(𝑛))
𝑓(1) + 𝑓(2) + 𝑓(3) + ⋯ + 𝑓(𝑛)
2
= 2550
Summation Notation
– 5 + 2 + 9 + 16 +··· + 268
𝑢𝑝𝑝𝑒𝑟 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟
𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎
40
7𝑛 − 12
𝑙𝑜𝑤𝑒𝑠𝑡 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟
7 + 11 + 15 + ··· + 207
51
4𝑛 + 3
Step 1
𝑛=1
𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎
𝑓(𝑛) =7 + (𝑛 − 1) ⋅ 4
= 7 + 4𝑛 − 4
= 4𝑛 + 3
𝑛=1
Arithmetic Series
Explicit Formula = Linear
Slope = Common Difference
Step 2
𝑢𝑝𝑝𝑒𝑟 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟
207 = 4𝑛 + 3
204 = 4𝑛
51 = 𝑛
Find
the
sum
of
a
Finite
Arithmetic
Series
4
40
𝑛3
(3𝑛 − 8)
𝑛=1
𝑛=1
𝑛
𝑆𝑛 = (𝑓(1) + 𝑓(𝑛))
2
Step 1: calculate the value of the first term
𝑓(1) = 3 1 − 8 = −5
Step 2: calculate the value of the last term
𝑓(40) = 3 40 − 8 = 112
Step 3: calculate the sum of all 40 terms
𝑆40
40
=
(−5 + 112) = 𝟐𝟏𝟒𝟎
2
NOT linear!!
𝑓(1) =13 = 1
𝑓(2) = 23 = 8
𝑓(3) = 33 = 27
𝑓(4) =43 = 64
100
A company pays a $10,000 bonus to salespeople at the
end of their first 50 weeks if they make 10 sales in
their first week, and then improve their sales numbers
by two each week thereafter. One salesperson
qualified for the bonus with the minimum possible
number of sales.
How many sales did the salesperson make in week 50?
𝑓(50) = 10 + (50 − 1) ⋅ 2 = 108
How many sales did the salesperson make in all 50 weeks?
𝑆50
50
(10 + 108) = 2950
=
2
p. 591: 9-45 mult. of 3
and 47