A sequence is generated by the recurrence relation Ur+1 = 1/Ur -3. Given that U2 = U1 find the possible of U1
U1=U1
U2=U1
U3=U3
U4=U4
for r>3, we have
U5 = 1/U1 = 1/U1
U6 = 1/U2 = 1/U1
U7 = 1/U3
Without some info about U3 and U4, I don't see what we can deduce.
To find the possible values of U1, we need to substitute the given condition into the recurrence relation.
Given: U2 = U1
Substituting U2 = U1 into the recurrence relation Ur+1 = 1/Ur -3, we get:
U1+1 = 1/U1 -3
Simplifying further:
U2 = 1/U1 -3
Since U2 = U1, we can substitute U1 for U2:
U1 = 1/U1 -3
Now, we can solve this equation for possible values of U1. Let's rearrange the equation:
U1 - 1/U1 = 3
To simplify further, let's multiply through by U1:
U1^2 - 1 = 3U1
Rearranging again:
U1^2 - 3U1 - 1 = 0
This is a quadratic equation. We can solve it using the quadratic formula:
U1 = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -3, and c = -1. Plugging these values into the quadratic formula, we get:
U1 = (-(-3) ± sqrt((-3)^2 - 4(1)(-1))) / (2(1))
Simplifying:
U1 = (3 ± sqrt(9 + 4)) / 2
U1 = (3 ± sqrt(13)) / 2
So, the possible values of U1 are given by:
U1 = (3 + sqrt(13)) / 2, (3 - sqrt(13)) / 2
To find the possible values of U1, we can start by substituting the given information U2 = U1 into the recurrence relation.
The recurrence relation is given by:
Ur+1 = 1/Ur - 3
Substituting U2 = U1 into the relation, we get:
U3 = 1/U2 - 3
Since U2 = U1, we can write this as:
U3 = 1/U1 - 3
Now, we need to use the given information that U2 = U1 to find the possible values of U1. Let's solve for U1:
U3 = 1/U1 - 3
Adding 3 to both sides of the equation:
U3 + 3 = 1/U1
Taking the reciprocal of both sides:
1/(U3 + 3) = U1
So, the possible values of U1 for which U2 = U1 are 1/(U3 + 3), where U3 is any valid value of the next term in the sequence.