aritmetic series?

Question

A sequence of therms {Un} is defined for n≥1, by the recurrence relation
Un-2=2KUn+1 + 15Un
where k is a constant. given than Un=1 and U2=-2

(a) find an expression, in terms of k, for U3

(b) hence find an expression, in terms of k, for U4

given also that U4= -38

(c) find the possible values of k.

Answer 1

Hi,
are you sure you copied the question correctly?

Instead of
Un-2=2KUn+1 + 15Un
where k is a constant. given than Un=1 and U2=-2


it would make more sense if it was, for example
U_{n+2}=2KU_{n+1} + 15U_n
where k is a constant. given than U_1=1 and U_2=-2

then, for n=1, you have
U_{1+2}=2KU_{1+1} + 15U_1
U_3 = 2KU_2 + 15U_1 (U_1=1 and U_2=-2)
U_3 = -4K + 15

and for n=2
U_{2+2}=2KU_{2+1} + 15U_2
U_4 = 2KU_3 + 15U_2 (U_2=-2 and U_3 = -4K + 15)
U_4 = 2K(-4K + 15) - 30
U_4 = -8K^2 + 30K -30

If U_4 = - 38, then
-8K^2 + 30K -30= - 38 (rearrange)
-8K^2 + 30K + 8 = 0

use the quadratic formula [sqrt means square root]:

K = { - 30 + - sqrt[30^2 - 4*(-8)*8] } / {2*(-8)}
K = {-30 + - sqrt(1156)} / (-16)
K = {-30 + - 34} / (-16)

so either K = {-30 + 34} / (-16) = 4/(-16) = -1/4

or K = {-30 - 34} / (-16) = (-64)/(-16) = 4





Hope this helps.