Sequences Day 6 Happy Thursday!!

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.

Questions on the Homework!?

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.

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.

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. 24 12 =2 ) So the nth term can be found using, 𝑎 𝑛 =3 2 𝑛−1

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

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

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

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

Finding terms given any two terms The third term of a geometric sequence is 63 4 and the sixth term is 1701 32 . 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

Finding terms given any two terms Given what we know about 𝑎 3 and 𝑎 6 63 4 =𝑎 𝑟 2 1701 32 =𝑎 𝑟 5 We can solve this system by dividing. 𝑎 𝑟 5 𝑎 𝑟 2 = 1701 32 63 4 → 𝑟 3 = 27 8 𝑟= 3 2

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

Fifth term 𝑎 𝑛 =7 3 2 𝑛−1 𝑎 5 =7 3 2 5−1 𝑎 5 =7 3 2 4 = 567 16

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 =3888 𝑎 3 =108

Homework: Page 839: #’s 1-10,17,18,20,22