Chapter 9-3 Arithmetic Sequences and Series (Day 2)
Obj: To find the sums of an Arithmetic Series.
Who uses this? – You can use arithmetic sequences to predict costs. See number 36 in homework
Consider the Sum of the following arithmetic sequence 1, 2, 3, 4, 5, 6, 7
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
The sum of the terms of a sequence is called a series. The series
above is an Arithmetic Series. (Why…?)
Arithmetic Sequence
vs.
Arithmetic Series
4, 7, 10, 13, 16
4 + 7 + 10 + 13 + 16
Sum of an Arithmetic Series - The Sum (𝑆𝑛 )of the first 𝑛 terms is:
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
Sum
# of Terms
1𝑠𝑡 Term
𝑛𝑡ℎ Term (end)
𝑛
*Use your Calculator and the formula 𝑠𝑛 = (𝑎1 + 𝑎𝑛 )
2
Ex 1. Find the sum of the first 50 positive even integers
𝑛
𝑆𝑛 = 2 (𝑎1 + 𝑎𝑛 )
𝑆50 =
Sub in the values for 𝑛, 𝑎1, & 𝑎𝑛
50
(2 + 100)
2
𝑆50 = 25(102) = 2250
Ex 2. Find the sum of the whole numbers from 1 to 100
What if we are looking for the sum of the following Arithmetic Series?
𝑛
𝑆40 𝑓𝑜𝑟 70 + 49 + 28 + 7 + ⋯
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
Step 1 – find 𝑎𝑛 using 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
Step 2 – plug the values for 𝑆𝑛, 𝑛, 𝑎1
OR…
𝑛
&
𝑎𝑛 into 𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
𝑛
𝑆𝑛 = (𝑎1 + 𝑎𝑛 )
2
𝑎𝑛 = 𝑎1 + (𝑛 − 1)
Sub in
𝑛
𝑆𝑛 = {𝑎1 + [𝑎1 + (𝑛 − 1)]}
2
which simplifies to..
𝑛
𝑆𝑛 = 2 [2𝑎1 + (𝑛 − 1)𝑑]
**This formula can be used when you do not know the value of the last term.
Now find the same sum using the above formula
𝑆40 𝑓𝑜𝑟 70 + 49 + 28 + 7 + ⋯
You try … Find
𝑆15 𝑓𝑜𝑟 25 + 12 + (−1) + (−14) + ⋯
Sigma Notation -
(or Summation Notation)
2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 =
Can be expressed as:
It is read “the sum of 2n as n
increases from 1 to 10”
Ex 1. Find
Need to find the 1st and 12th terms
Homework Ch 9-3 (day 2) pg 648, 17-19, 33-36