The sequence x_1, x_2, x_3, . . ., has the property that x_n = x_{n - 1} + x_{n - 2} for all n \ge 3. If x_{11} - x_1 = 99, then determine x_6.

Steve, you made a mistake for x_4.

well, did you fix it? Not sure what you mean, since I had x11-x1 = 99 and the sequence

3,0,3,3,6,9,15,24,39,63,102

has Xn = X(n-1)+X(n-2) for n>2

everything got confusing after that

You might want to study up on the Fibonacci series. That's what the Fn values were.

To determine the value of x_6, we need to understand the given sequence and use the information provided.

The sequence x_1, x_2, x_3, ..., follows a recursive relationship. In this case, each term x_n is equal to the sum of the two preceding terms, x_{n-1} and x_{n-2}.

Let's write out the first few terms of the sequence:
x_1, x_2, x_3, x_4, x_5, x_6, ...

Given that x_{11} - x_1 = 99, we can write the equation as:

x_11 - x_1 = x_{10} + x_9 - x_1 = x_9 + (x_8 + x_7) - x_1 = x_9 + x_8 + x_7 - x_1 = 99

We can see that x_1 cancels out, leaving us with:

x_9 + x_8 + x_7 = 99

Using the recursive relationship, we can further simplify this equation.

Since x_6 = x_5 + x_4, we can express x_5 as x_6 - x_4. Similarly, x_4 can be expressed as x_5 - x_3, and x_3 can be expressed as x_4 - x_2.

Replacing these expressions in the equation, we have:

(x_6 - x_4) + (x_5 - x_3) + x_7 = 99

Simplifying, we get:

x_6 + (x_7 - x_3) = 99

Now, let's reconsider the given information that the sequence satisfies the recursive relationship:

x_n = x_{n-1} + x_{n-2}

Knowing this, we can substitute x_7 as x_6 + x_5, and x_3 as x_2 + x_1:

x_6 + (x_6 + x_5 - x_2 - x_1) = 99

Since we are given x_1 as x_1, we can substitute the value:

x_6 + (x_6 + x_5 - x_2 - x_1) = 99

Now, we know that x_{11} - x_1 = 99, and x_6 + x_7 - x_3 = 99.

This means that x_{11} - x_1 = x_6 + x_7 - x_3.

Substituting, we get:

x_6 + x_7 - x_3 = x_6 + x_7 - x_3

Therefore, there is no mistake in x_4. We have successfully derived the equation x_{11} - x_1 = x_6 + x_7 - x_3.

To determine the value of x_6, we need additional information about the sequence or the specific values of some terms. Without further information, we cannot determine the value of x_6.