SOLUTION: Can someone help me? A sequence {g n} is defined recursively as g 1=72, g n=-2/3*g n-1 , for n ≥2 (note: here "n" & "1"& "n-1" after "g" should be a smaller letter, I can

Algebra ->  Sequences-and-series -> SOLUTION: Can someone help me? A sequence {g n} is defined recursively as g 1=72, g n=-2/3*g n-1 , for n ≥2 (note: here "n" & "1"& "n-1" after "g" should be a smaller letter, I can      Log On


   

Question 1090885: Can someone help me?
A sequence {g n} is defined recursively as g 1=72, g n=-2/3*g n-1 , for n ≥2
(note: here "n" & "1"& "n-1" after "g" should be a smaller letter, I cannot put it here as smaller)
a. Determine the values of g 2, g 3, and g 4
b. Write an explicit formula for g n
Thank you.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I believe you meant g%5B1%5D=72 and g%5Bn%5D=-%282%2F3%29%2Ag%5Bn-1%5D .

a) system%28g%5Bn%5D=-%282%2F3%29%2Ag%5Bn-1%5D%2Cg%5B1%5D=72%29 -->
g%5B2%5D=-%282%2F3%29%2A72
g%5B2%5D=-2%2A72%2F3
highlight%28g%5B2%5D=-48%29
g%5B3%5D=-%282%2F3%29%2A%28-48%29
g%5B3%5D=2%2A48%2F3
highlight%28g%5B3%5D=32%29
g%5B4%5D=-%282%2F3%29%2A32
g%5B4%5D=-2%2A32%2F3
highlight%28g%5B4%5D=-64%2F3%29

b) g%5Bn%5D=g%5B1%5D%2Ar%5E%28n-1%29 is the general explicit formula for a geometric sequence with first term g%5B1%5D and common ratio r .
For the sequence in this problem,
highlight%28g%5Bn%5D=72%2A%28-2%2F3%29%5E%28n-1%29%29 ,
That is "an explicit formula" for g%5Bn%5D .
As 72=8%2A9=2%5E3%2A3%5E2 , the formula could be "simplified"
g%5Bn%5D=72%2A%28-2%2F3%29%5E%28n-1%29
g%5Bn%5D=2%5E3%2A3%5E2%2A%28-1%29%5E%28n-1%29%2A2%5E%28n-1%29%2F3%5E%28n-1%29
highlight%28g%5Bn%5D=%28-1%29%5E%28n-1%29%2A2%5E%28n%2B2%29%2F3%5E%28n-3%29%29
That last equation is also "an explicit formula" for g%5Bn%5D .
As a formula it may not look simple,
but it would help if you had to calculate several terms,
because it would simplify calculations like
g%5B4%5D=%28-1%29%5E3%2A2%5E6%2F3%5E1=-64%2F3 , and
g%5B7%5D=%28-1%29%5E6%2A2%5E9%2F3%5E4=512%2F81 .