Discover the Constant Values of a Math Sequence | Need Help with Math

[preet]

A sequence $$t_{1}$$,$$t_{2}$$,...$$t_{n},...$$ is defined as follows:

$$t_{1}=14$$
$$t_{k}=24-5t_{k-1}$$ for each k <= 2

For every positive integer $$n$$, $$t_{n}$$ can be expressed as $$t_{n}=p*q^n+r$$ where p, q and r are constants. The value of p, q and r is?

I found out $$t2=-46$$ and $$t3=254$$ and wrote out

$$14=p*q+r\\$$
$$-46=p*q^2+r\\$$
$$254=p*q^3+r$$

I don't know what to do from here... is what I did so far even correct? TiA.

 

[arildno]

First of all, the recurrence relation ought to be:
$$t_{k}=24-5t_{k-1},k>=2$$
Now, for arbitrary k, you have been given:
$$t_{k}=pq^{k}+r,t_{k-1}=pq^{k-1}+r$$
Put this into the recurrence relation, and rearrange a bit:
$$(q+5)pq^{k-1}=24-6r$$
This equality must hold for EVERY k>=2. What does that suggest to you?

 

[preet]

I don't really get what's happening...

 

[HallsofIvy]

Well, the first thing that happened is that you had a "misprint". You had written "for each k<= 2" when you clearly meant "for each k>= 2". That was what arildno was talking about.

THEN his point was that, if you know t_n= p qⁿ+ r, you can put that into the recursion formula. Replacing n by n-1 gives t_n-1= p q^n-1+ r so the recursion formula says p qⁿ+ r= 24-5(p q^n-1+ r). That equation is true for all n and has 3 unknown numbers in it so you can evaluate it for 3 values of n (2, 3, 4 will do) to get 3 equations for those 3 unknowns.
That's exactly what you did so, yes, what you have done so far is correct.

Now, to solve those equations:

An obvious thing to do is subtract the first equation from the second so that you get
60= pq- pq²=pq(1-q).

Subtract the second equation from the third to get 300= pq³- pq²= pq²(1-q).

Now, from the first equation, p= 60/(q(1-q)) so the second equation is
300= 60 (q²(1-q))/(q(1-q)). That is, 300= 60 q or q= 5.
Now work back to solve for p and r.

 

[preet]

I got it... thanks for your help!

...but now I have another question that I'm stuck on. I can't even begin to do anything because I have no idea how to find out the answer... so some hints or something would be appreciated.

A circle that has a radius of 3 is tangent to both the positive x-axis and the positive y-axis. Another circle with a radius of 1 is tangent to the bigger circle and to the positive x-axis. There is a line tangent to both circles. Where does this line cross the y-axis?

I had to change the question around a bit... there's a picture associated that makes it eaiser to imagine but its not that hard to draw out... and the line isn't the x-axis!

 

[xanthym]

Refer to the following diagram:
http://img110.exs.cx/img110/7287/tangentcircles4lh.jpg
From the RED Right-Triangle construct, determine the coordinates of Points A, B, C, and D. The equation of Line AC through points A and C can then be found, and thus the equation of Line ED through Point E and Normal to Line AC at Point D can be determined. The Y-intercept of Line ED is the solution to your problem.

 

[preet]

Thanks for the help... I have another (simple) question: How do I evaluate (3^2000+3^-2000)^2-(3^2000-3^-2000)^2

 

[Curious3141]

Better to put different questions into different threads.

For the last one, use $$a^2 - b^2 = (a+b)(a-b)$$ and see what you get.

 

[preet]

Ill do that in the future... I just don't know how to simplify (do I even simplify?) something like 3^2000+3^2000...

this is the solution... but I don't understand where 2.3 came from and how it becomes 4.3, etc...

http://img77.exs.cx/img77/8510/dme9zz.gif

 

[xanthym]

Ill do that in the future... I just don't know how to simplify (do I even simplify?) something like 3^2000+3^2000...

this is the solution... but I don't understand where 2.3 came from and how it becomes 4.3, etc...

http://img77.exs.cx/img77/8510/dme9zz.gif

{2.3²⁰⁰⁰} should be interpreted {2*3²⁰⁰⁰}
{4.3⁰} should be interpreted {4*3⁰}


~~

 

[preet]

wicked... it makes sense now =)