How to Solve the Difference Equation yn+1=(n+1)/(n+2) yn?

[Success]

Solve the difference equation y_n+1=(n+1)/(n+2) y_n in terms of the initial value y₀.

 

[tiny-tim]

Hi Success!

Show us what you've tried, and where you're stuck, and then we'll know how to help!

(common-sense should solve this)

 

[Success]

I really don't even know how to start. I guess you should begin with yn=1/(n+1) y0.

 

[jhosamelly]

Use Quotient Rule

\frac{d}{dt}\left(\frac{u}{v}\right) = \frac{vu'-uv'}{v^2}

You should be able to solve this.

 

[tiny-tim]

I really don't even know how to start. I guess you should begin with yn=1/(n+1) y0.

do you mean y₁ = 1/(2) y₀ ?

yes you could start with that, then find y₂, then y₃, …

and see if you can spot a pattern ​

 

[HallsofIvy]

I really don't even know how to start. I guess you should begin with yn=1/(n+1) y0.

IF that is true then it is the answer to your problem! If you can start writing down the answer, you surely don't need our help! How did you get that?

Poor, not so brilliant people like me might start by writing out a few values and looking for a pattern. If y_{n+1}= ((n+1)/(n+2))y_n, then y_1= ((0+1)/(0+2))y_0= y_0/2, y_2= ((1+1)/(1+2))y_1= 2y_1/3= 2(y_0/2)/3= y_0/3, y_3= ((2+1)/(2+2)y_2= (3/4)y_2= (3/4)(y_0/3)= y_0/4, y_4= ((3+1)/(3+2)y_3= (4/5)y_3= (4/5)(y_0/4)= y_0/5...

Do you think you can make a guess now? To be complete, you should then prove that your guess is correct, by induction, say.

 

[Success]

jhosamelly, I got 3/(n+2)^2 by quotient rule. How is that the answer?

 

[Success]

Thanks everyone.

 

[HallsofIvy]

jhosamelly, I got 3/(n+2)^2 by quotient rule. How is that the answer?

This problem has nothing to do with Calculus, the derivative, or the quotient rule.

 

[jhosamelly]

This problem has nothing to do with Calculus, the derivative, or the quotient rule.

Ow, sorry, I thought your title is "solve the DIFFERENTIAL equation" .

 

[tiny-tim]

spot the difference!

 

[HallsofIvy]

Ow, sorry, I thought your title is "solve the DIFFERENTIAL equation" .

But what you wrote had nothing to do with solving a differential equation either.