The sequence < un > is defined by the recurrence
Un+1 = 3Un+1\5Un+3
initial condition of u1 = 1:
Need to show un in terms of Fibonacci / Lucas numbers
To show the terms of the sequence <un> in terms of Fibonacci or Lucas numbers, we need to find a closed-form expression for un in terms of these special sequences.
First, let's rewrite the given recurrence relation:
Un+1 = (3Un + 1) / (5Un + 3)
We can start by multiplying both sides of the equation by (5Un + 3):
Un+1 * (5Un + 3) = 3Un + 1
Expanding the left side:
5Un+1 + 3Un+1 = 3Un + 1
Combining like terms:
8Un+1 = 3Un + 1
Now, let's rearrange the equation to solve for Un+1:
Un+1 = (3Un + 1) / 8
This gives us the expression for Un+1 in terms of Un.
To express Un in terms of Fibonacci or Lucas numbers, we can use the following formulas:
For Fibonacci numbers:
Fn = (phi^n - (-phi)^(-n)) / sqrt(5)
For Lucas numbers:
Ln = phi^n + (-phi)^(-n)
where phi is the golden ratio, approximately 1.618.
To find an expression for Un in terms of Fibonacci or Lucas numbers, we need to derive a relationship between Un and Un+1.
Notice that Un+1 = (3Un + 1) / 8 can be written as:
Un+1 = 3/8 * Un + 1/8
Now, if we assume Un = aFn + bLn for some constants a and b, we can substitute this into the equation:
aFn+1 + bLn+1 = 3/8 * (aFn + bLn) + 1/8
Expanding both sides and regrouping terms:
a(Fn+1 - 3/8Fn) + b(Ln+1 - 3/8Ln) = 1/8
To match the left-hand side of the equation with the right-hand side, we must have:
a = 1/8
b = 0
Therefore, Un = (1/8)Fn, where Fn is a Fibonacci number.
So, the terms of the sequence <un> can be expressed in terms of Fibonacci numbers as Un = (1/8)Fn.
Note: The above analysis assumes that the sequence <un> converges and has a closed-form solution in terms of Fibonacci or Lucas numbers. This may not always be the case, and additional analysis or calculations might be necessary to conclusively determine the relationship.