Geometric Progression: Given a_0=12, a_5=24, and a_10=48, find expression for n-th term

J

Jeanlou

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Hi great if you could please help me solve this:

Given:
a0=12
a5=24
a10=48

r = common ratio (i.e. 2)
n = number of notes in scale (i.e. 5)
a0 = fundamental frequency (i.e. 12)

What is the equation to find ath value:
i.e. a1, a2, a3

Thanks
Jean-Lou
 
a0 = a0r0,

a1 = a0r1,

a2 = a0r2,

a3 = a0r3,

a4 = a0r4,

a5 = a0r5,.......

what do you get for

a5 / a0 = ?

continue....
 
I think you are saying that these are three terms in a geometric progression [MATH]a_i[/MATH] that doubles every 5th term, and you want the ith value, [MATH]a_i[/MATH]. The "r" for this sequence, of course, is not 2. Rather, [MATH]r^5 = 2[/MATH]. So what is r? What is the formula?
 
Y
Dr.Peterson said:
I think you are saying that these are three terms in a geometric progression [MATH]a_i[/MATH] that doubles every 5th term, and you want the ith value, [MATH]a_i[/MATH]. The "r" for this sequence, of course, is not 2. Rather, [MATH]r^5 = 2[/MATH]. So what is r? What is the formula?
Click to expand...
Yes please!
 
Do mean than a0 multiply by r=[MATH]\sqrt[5]{2}[/MATH] would give me a1?
 
Jeanlou said:
Do mean than a0 multiply by r=[MATH]\sqrt[5]{2}[/MATH] would give me a1?
Click to expand...
I answered exactly the question as you wrote it.
You posted: " I need to know the equation to get r".
 
Jeanlou said:
Do mean than a0 multiply by r=[MATH]\sqrt[5]{2}[/MATH] would give me a1?
Click to expand...
Yes, that's what [MATH]r[/MATH] means in this context. The formula for the sequence is [MATH]a_i = a_0\cdot r^i[/MATH], and in particular [MATH]a_1 = a_0\cdot r[/MATH].
 
Given:
a0=12
a5=24
a10=48

Initial term = a0
n= 5 (a0, a1, a2, a3, a4)
r = a5/a0 = 2

To find the ith term: ai = a0 • ri
To find r:
rn = 2
r = 21/n

so
ai = a0 • 2i/n

Isn't it?
 
You're using r as you defined it, and then as it is used within the sequence you are asking about; so you have used r to mean two different things.

But, keeping those distinct, your final line is correct (with n=5). And, of course, a0=12.
 
Yes that is correct. Usually a geometric progression is written as \(\displaystyle g_n=a\cdot r^n\) where \(\displaystyle a\) is the first term and \(\displaystyle r\) is the common ratio.
So that \(\displaystyle g_0=a\cdot r^0=a\) or the first term. If we know any other term then we know the value of \(\displaystyle r\).
If \(\displaystyle a=4\) and \(\displaystyle g_{10}=200\) then because \(\displaystyle 200=4\cdot 50\) then \(\displaystyle r^{10}=50\) & \(\displaystyle r=\sqrt[10]{50}\).

Thus \(\displaystyle g_n=4\cdot (\sqrt[10]{50})^n\)

 
At use on this website, hope writing make sense, but at least it's working great!!! Thanks a lot guys!
 
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