Sequences and series help / recurrence relation

[tweety1234]

Homework Statement

A sequence of terms [tex] U_{k} [/tex] is defined by [tex] K \geq [/tex] by the recurrence relation [tex] U_{k+2} = U_{k+1} - pU_{k} [/tex] where P is a constant Given that [tex] U_{1} =2 [/tex] and [tex] U_{2} = 4 [/tex]

a) find an expression in terms of p for [tex] U_{3} [/tex]

b) hence find an expression in terms of p for [tex] U_{4} [/tex]

given also that [tex] U_{4} [/tex] is twice the value of [tex] U_{3} [/tex]
c) find the value of p

The Attempt at a Solution

for question a i just subsititue k=1 and i get [tex] U_{3} = 4 - 2p [/tex] and for B i substituted k=2 and the expression i got is [tex] U_{4} = 4-6p [/tex]

what i am really stuck on is how to work out the value of 'p' ?

can anyone please show me ?

thanks!

 

[HallsofIvy]

Homework Statement

A sequence of terms [tex] U_{k} [/tex] is defined by [tex] K \geq [/tex] by the recurrence relation [tex] U_{k+2} = U_{k+1} - pU_{k} [/tex] where P is a constant Given that [tex] U_{1} =2 [/tex] and [tex] U_{2} = 4 [/tex]

a) find an expression in terms of p for [tex] U_{3} [/tex]

b) hence find an expression in terms of p for [tex] U_{4} [/tex]

given also that [tex] U_{4} [/tex] is twice the value of [tex] U_{3} [/tex]
c) find the value of p

The Attempt at a Solution

for question a i just subsititue k=1 and i get [tex] U_{3} = 4 - 2p [/tex] and for B i substituted k=2 and the expression i got is [tex] U_{4} = 4-6p [/tex]

what i am really stuck on is how to work out the value of 'p' ?

can anyone please show me ?

thanks!

Excellent! You have done exactly what you should have done. Now use that last condition, U₄ is twice the value of U₃ or U₄= 3U₃ with the U₃ and U₄ you have and solve for p.

 

[Architeuthis Dux]

I would begin by looking at the given information and trying to understand the problem at hand. From the given recurrence relation, we can see that the sequence is defined by taking the previous two terms and subtracting the product of the constant p and the first term. This means that each term in the sequence is dependent on the previous two terms and the value of p.

To find the value of p, we can use the given information that U4 is twice the value of U3. This means that U4 = 2U3. We can substitute our expressions from parts a and b into this equation to get:

4 - 6p = 2(4 - 2p)

Solving for p, we get p = 1. This means that the value of p is 1 in this particular sequence.

In general, to find the value of p in a sequence defined by a recurrence relation, we would need more information. Without additional information, we cannot determine the value of p. It is also possible that there are multiple values of p that could satisfy the given conditions. In this case, we would need more information to narrow down the possible values of p.