A sequence of numbers U1,U2,U3,... satisfies the relation:(3n-2)Un+1=(3n+1)Un for all in nEZ.If U1=1,find
(i)U3 and U4
U1 = 1
(3*1-2)U2 = (3*1+1)U1
U2 = 4U1
U2 = 4
Now do that process twice more to find U3 and U4
U3=7, and U4=10
Let (Un)be a Sequence satisfies that limit (Un+1/Un)=0 Determine the limit of (Un).
Oh, numbers and their relationships! It's like they're in a complicated love affair. Let's see if we can make some sense out of it.
Given that (3n-2)Un+1=(3n+1)Un, we can start finding the values step by step.
(i) To find U3, we can use the equation:
(3(1) - 2)U2 = (3(1) + 1)U1
U2 = (4/1)U1
U2 = 4(1)
U2 = 4
Now, let's move on to U4, shall we?
(3(2) - 2)U3 = (3(2) + 1)U2
4U3 = 7(4)
U3 = 7
Therefore, U3 = 7 and U4 = 4.
Oh, the wonders of numbers and their intricate connections! Keep those mathematical gears turning, my friend!
To find U3 and U4, we can use the given relation and substitute the values of n.
Let's start with finding U3:
Plugging in n = 1 into the given relation:
(3(1) - 2)U2 = (3(1) + 1)U1
U2 = 4U1 - 3
Now, let's find U4:
Plugging in n = 2 into the given relation:
(3(2) - 2)U3 = (3(2) + 1)U2
4U3 - 2U3 = 9U2
However, to find U4, we need to know the value of U2.
To find U2, we can use the given relation and substitute the value of n:
Plugging in n = 1 into the given relation:
(3(1) - 2)U2 = (3(1) + 1)U1
U2 = 4U1 - 3
U2 = 4(1) - 3
U2 = 1
Now that we know U2 = 1, we can substitute it back into the equation for U4:
4U3 - 2U3 = 9U2
4U3 - 2U3 = 9(1)
2U3 = 9
U3 = 9/2
Therefore, U3 = 9/2 and U4 is still unknown since we need U2 to compute it.