1. Sequences and Series Day 7
Good afternoon Happy Tuesday!

2. Warm-Up
How can we identify if a sequence is arithmetic, geometric or neither?
Given 𝑎 𝑛 = 𝑛−1 what is the first term in the sequence?
3) Find the sum: 𝑘=3 6 𝑘 2

3. Questions on the homework?

4. Partial Sum of Geometric Sequences
Just like arithmetic sequences we are going use partial sums with geometric sequences(aka geometric series) .
For the geometric sequence 𝑎, 𝑎𝑟, 𝑎 𝑟 2 ,𝑎 𝑟 3 ,… 𝑎𝑟 𝑛−1 ,…the nth partial sum is
𝑆 𝑛 = 𝑘=1 𝑛 𝑎 𝑟 𝑘−1 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +…+𝑎 𝑟 𝑛−1

5. Partial Sum formula
With out going through the proof: We need to multiply 𝑆 𝑛 by 𝑟 and subtract that from 𝑆 𝑛 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 → 𝑆 𝑛 1−𝑟 =𝑎 1− 𝑟 𝑛 So 𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟 , 𝑟≠1

6. Summary
The formula to find the nth partial sum of a geometric sequence is 𝑆 𝑛 =𝑎 1− 𝑟 𝑛 1−𝑟

7. Practice
Find the sum of the first five terms of the geometric sequence: 1,0.7,0.49,0.343 𝑎=1, 𝑟=0.7 Using the formula for 𝑆 𝑛 and n=5 We get 𝑆 5 =1∗ 1− −0.7 = So the sum of the first five terms is

8. Practice
Find the sum 𝑘=1 5 7 − 2 3 𝑘 We need find a, r and n. Then use the formula, keep in fraction form.

9. Answer Find the sum 𝑘=1 5 7 − 2 3 𝑘 𝑛=5, 𝑟=− 2 3 , 𝑎=7∗− 2 3 =− 𝟏𝟒 𝟑
𝑛=5, 𝑟=− 2 3 , 𝑎=7∗− 2 3 =− 𝟏𝟒 𝟑
Why is a=− 14 3 and not 7 ?
since k is 1, we start there. If k=0 then a=7
Formula 𝑆 𝑛 =− ∗ 1− − −− 2 3 =
− 14 3 ∗ =− =−3.1687

10. You Try
Are the following geometric if so what is the common ratio?
3,6,12,24…
2,4,6,8…
1 2 , 1 6 , 1 18 , 1 54 …
Find the partial sum 𝑆 𝑛 of the geometric sequence that satisfies the given conditions
4) 𝑎=3, 𝑟=4, 𝑛=6
5) 𝑎= 5 6 , 𝑟= 1 6 , 𝑛=4

11. Solutions
1) Yes, r=2 2) No, arithmetic 3) Yes , r= 1/3 4) 𝑆 6 =3 1− 4 6 1−4 =4095 5) 𝑆 4 = 5 6 1− − 1 6 =

12. Homework Tonight
Page 840:#’s 23-25,27-30, 31,33, 35 (hint for 25 you will have to find 𝑎 and 𝑟 first)

13. Infinite Series