Sequences and series help / recurrence relation

[tweety1234]

Homework Statement

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

a) find an expression in terms of p for $$U_{3}$$

b) hence find an expression in terms of p for $$U_{4}$$

given also that $$U_{4}$$ is twice the value of $$U_{3}$$
c) find the value of p

The Attempt at a Solution

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

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 $$U_{k}$$ is defined by $$K \geq$$ by the recurrence relation $$U_{k+2} = U_{k+1} - pU_{k}$$ where P is a constant Given that $$U_{1} =2$$ and $$U_{2} = 4$$

a) find an expression in terms of p for $$U_{3}$$

b) hence find an expression in terms of p for $$U_{4}$$

given also that $$U_{4}$$ is twice the value of $$U_{3}$$
c) find the value of p

The Attempt at a Solution

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

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.