Math sequence question??

Question

consider the seq. (Un) definced by Uo=o and U(n+1)=(2Un+3)/(Un+4):
a) Calculate U1,U2,U3.
b)Let (Vn) be the sequence defined by: Vn=(Un-1)/(Un+3).

-Show that (Vn+1)/(Vn) is independant of n.

-deduce the nature of(Vn).
Thank you for your time...

Answer 1

It is already said by the former answerers that the values of U1=3/4 ; U2= 18/19, U3=93/94 are true. this series is neither Arithmetic nor Geometric sequence since it has neither common difference nor common ratio. Hence the given equation is a function whose value is approaching 1.
Now, answer to the question b) is the ratio (Vn+1)/(Vn) should be equal to a constant to be independent of n . Since given series is not a sequence having constant CR or CD given ratio cannot be independent of n. Moreover, (a+1)/b is not equal to (a+2)/(b+1)
Hence the given ratio is not independent of n.

Answer 2

Continuing from where the last answer finished.
Vo = (0 - 1)/(0 + 3) = -1/3
V1 = (3/4 - 1)/(3/4 + 3) = -1/15
V2 = (18/19 - 1)/(18/19 + 3) = - 1/75
V3 = (93/94 - 1)/(93/94 + 3) = -1/375
This indicates that V(n + 1)/V(n) = 1/5 and that V(n) is a geometric sequence. However proving this is harder. I'll come back with a later edit when I've had a go at it.

Later edit. It's actually quite easy. Just plug the definition of
U(n + 1) into V(n + 1)/V(n) and the whole thing cancels to 1/5 quite nicely.

Answer 3

a) U1 = U(0+1) so n=0 and we know that Uo=0
U(0+1) = (2Uo+3)/(Uo+4)
= (2(0)+3)/((0)+4) = (3)/(4) = 3/4

U2 = U(1+1) so n = 1 and we know that U1= 3/4
U(1+1) = (2U1+3)/(U1+4)
= (2(3/4)+3)/(3/4 +4) = (9/2)/(19/4) = 18/19

U3 = U(2+1) so n=2 and we know that U2=18/19
U(2+1) = (2U2+3)/(U2+4)
= (2(18/19) +3)/((18/19)+4) = (93/19)/(94/19) = 93/94