1. Sequences Day 6
Happy Thursday!!

2. Warm-Up In an arithmetic sequence 𝑎 𝑛 =𝑎+ 𝑛−1 𝑑.
What do the variables a, d and n stand for?
Given a sequence how can we solve for d?
3) Write one formula for the sum of an arithmetic sequence.

3. Questions on the Homework!?

4. Geometric Sequences
Another simple way of generating a sequence is to start with a number 𝑎 and repeatedly multiply by a fixed nonzero constant 𝑟. Def: A geometric sequence is a sequence of the form 𝑎, 𝑎𝑟, 𝑎 𝑟 2 ,𝑎 𝑟 3 , 𝑎 𝑟 4 … 𝑎 is the first term, and r is common ratio of the sequence.

5. Geometric Sequences 𝒂 𝒏 =𝒂 𝒓 𝒏−𝟏
The nth term of a geometric sequence is given by
𝒂 𝒏 =𝒂 𝒓 𝒏−𝟏
Again r is called the common ratio because the ratio of any two consecutive terms of the sequence is r.

6. Example
If a=3 and r=2, the we have the geometric sequence, 3, 3∗2, 3∗ 2 2 , 3∗ 2 3 , 3∗ 2 4 ,… Or 3, 6, 12,24,48… Note the ratio between any two consecutive numbers is 2. (i.e =2 ) So the nth term can be found using, 𝑎 𝑛 =3 2 𝑛−1

7. Example Find the nth term of the Sequence: 2 , −10, 50, −250, 1250
𝑎=2 first term.
Find r , any term divided by the previous term
50 −10 =−5
The nth term : 𝑎 𝑛 =2 −5 𝑛−1

8. You try
Using the sequence 1, 1 3 , 1 9 , 1 27 , 1 81 Find the nth term. 𝑎=1 , 𝑟= 1 3 𝑎 𝑛 =1 1 3 𝑛−1

9. Finding terms
Find the 8th term of the geometric sequence 5,15,45… Find a and r. 𝑎=5, 𝑟= 15 5 =3 𝑎 𝑛 =5 3 𝑛−1 𝑝𝑙𝑢𝑔 8 𝑖𝑛 𝑓𝑜𝑟 𝑛 𝑎 8 =5 3 8−1 = 𝟓 𝟑 𝟕 =𝟏𝟎,𝟗𝟑𝟓

10. Good afternoon Happy Friday!
Recall the formula to find any term of a geometric sequence is: 𝑎 𝑛 =𝑎 𝑟 𝑛−1

11. Finding terms given any two terms
The third term of a geometric sequence is and the sixth term is Find the fifth term.
Need to find 𝑎 and 𝑟. Using 𝑎 𝑛 =𝑎 𝑟 𝑛−1
𝑎 3 =𝑎 𝑟 3−1 =𝑎 𝑟 2
𝑎 6 =𝑎 𝑟 6−1 =𝑎 𝑟 5

12. Finding terms given any two terms
Given what we know about 𝑎 3 and 𝑎 =𝑎 𝑟 =𝑎 𝑟 5 We can solve this system by dividing. 𝑎 𝑟 5 𝑎 𝑟 2 = → 𝑟 3 = 27 8 𝑟= 3 2

13. Finding terms given any two terms
Substituting for 𝑟 in the first equation 63 4 =𝑎 𝑟 =𝑎 → 63 4 =𝑎 9 4 𝑎=7 𝑎 𝑛 =7 3 2 𝑛−1

14. Fifth term
𝑎 𝑛 =7 3 2 𝑛−1 𝑎 5 = −1 𝑎 5 = =

15. Practice
Determine if the sequence is geometric if so find the nth term and 8th term.
−1,6,−36,216
−1 , 1, 4, 8
−2, −4,−8,−16
Given two terms find the nth term
4) 𝑎 1 =−2 𝑎 5 =−512
5) 𝑎 5 = 𝑎 3 =108

16. Homework:
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