Unit 8 Day 7 – Sums of Series
Algebra 2
Name:_______________
Today:
 You will learn how to find the sum of geometric and arithmetic sequences.
 At the end of class you will be able to evaluate the sum of a geometric or arithmetic series.
Warm-Up
5
1)
3 k
2
2) Find the 8th term of the arithmetic sequence whose common
k 2
difference is 4 and whose third term is -3.
Sums of Series
An Arithmetic Series is the expression formed by
adding terms of a finite arithmetic sequence.
Ex. 1 Gauss was given this detention assignment when he was in 3rd grade.
“Add all of the integers from 1 to 100 and then you can go home.”
He was able to do it in just 30 seconds. Can you? Try it!
1+2+3+4+5+…+97+98+99+100 = ??????
The Sum of an Arithmetic Series is:
1+2+3+4+5
Ex. 2 Find the sum of the arithmetic sequence without adding up all of the terms!
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
2, 8, 14, 20, …
Ex. 3 Find the sum of the 1st n terms of the arithmetic series:
n = 25
32
Ex. 4 Evaluate:
 2i  8
i 1
6
Ex. 5 Evaluate the sum of each geometric series:
 2(5)
i
The SUM of a Finite Geometric Series
is:
 1 rn 
S  a1 

 1 r 
i1
Without the formula:
where r is the common ratio and r  0 .
With the formula:
Ex. 6
 1
10   

 2
i 0
7
a)
Find the sum of the first 10 terms of the geometric series: 1  5  25  125  . . .
i
12
b)
 0.01(3)
i
i 0
****************************************************************************************
Infinite Geometric Series: goes on and on and on and on and on and on . . .
If the absolute value of the common ratio, r, is
_____ then the infinite series has __________.
Find the sum of all of the numbers in the infinite series:
1) 1  2  4  8  16  32  . . .
If the absolute value of the common ratio, r, is
_____ then this is the formula for the SUM OF
AN INFINITE SERIES:
S
a1
1 r
1 1 1 1 1
2) 1       . . .
2 4 8 16 32
Note: The sum gets closer and closer to _______. It will never quite reach it, or course, but we use it as the
sum anyway since it will get soooooooo very close.

3)
 4(0.6)
n 1
= _____+_____+_____+_____+. . .
n 1
a1  ______
r  ______
Can we find the sum?

1
4)   
n 0  2 
n

4
5)  20  
5
n 1
n 1
5
Review: 1. Expand and evaluate:
 (i
2
 2i )
i 2
2.
Write a rule for the nth term of the arithmetic sequence if a3  17 and d  4.
3.
Find the 12th term of the sequence: 108, 91, 74, 57,...
4.
Write a rule for the nth term of the geometric sequence if a3  36 and r  3 .
5.
Write the first three terms of each sequence.
a. an  2n  5
 1
b. an  2  
3
n
c. a1  3, an  2  3an 1
6.
During a free fall, a skydiver falls 16 feet in the first second, 48 feet in the 2nd second, and 80 feet in
the third second. If she continues to fall at this rate, how many feet will she fall during the 8th second?
7. Find the infinite sum of the following: a) ∑∞
𝑘=1
16 3 𝑘−1
9
( 2)
b) 𝑎1 = 1,
𝑟=
1
2