Suppose that the sequence x0, x1, x2... is defined by x0 = 1, x1 = 5, and xk+2 = 3xk+1+10xk for k≥0. Find a general formula for xk.
To find a general formula for xk, let's start by examining the given recursive relationship:
xk+2 = 3xk+1 + 10xk
We notice that the equation relates to the previous two terms, xk+2 and xk+1, in terms of multiples of xk. This hints at the possibility of a characteristic equation.
Let's assume a solution of the form: xk = r^k
Substituting this into the recursive equation, we have:
r^k+2 = 3r^k+1 + 10r^k
Dividing both sides by r^k, we get:
r^2 - 3r - 10 = 0
Now, we can solve this quadratic equation for r. Factoring it, we have:
(r - 5)(r + 2) = 0
So, we have two possible values for r: r = 5 and r = -2.
This gives us two linearly independent solutions: x1 = 5^k and x2 = (-2)^k.
To find the general formula, we use the superposition principle. We express xk as a linear combination of these two solutions:
xk = A * 5^k + B * (-2)^k
where A and B are constants to be determined.
To find A and B, we use the initial conditions given in the problem.
When k = 0, we have:
x0 = A * 5^0 + B * (-2)^0
1 = A + B
When k = 1, we have:
x1 = A * 5^1 + B * (-2)^1
5 = 5A - 2B
Now, we solve these two equations to find the values of A and B.
From the first equation, we can write: B = 1 - A
Substituting this into the second equation:
5 = 5A - 2(1 - A)
5 = 5A - 2 + 2A
7 = 7A
A = 1
Substituting this back into the first equation: 1 = 1 + B
B = 0
Therefore, the values of A and B are: A = 1 and B = 0.
Finally, we substitute these values back into our general solution:
xk = 1 * 5^k + 0 * (-2)^k
xk = 5^k
So, the general formula for xk is xk = 5^k.
To find a general formula for the given recursive sequence, we need to determine the pattern and derive a formula based on it.
Step 1: Generate a few terms of the sequence
Let's calculate the first few terms of the sequence using the given recursive relation:
x0 = 1
x1 = 5
x2 = 3x1 + 10x0 = 3(5) + 10(1) = 15 + 10 = 25
x3 = 3x2 + 10x1 = 3(25) + 10(5) = 75 + 50 = 125
x4 = 3x3 + 10x2 = 3(125) + 10(25) = 375 + 250 = 625
Step 2: Identify the pattern
By observing the calculated terms, we can notice a pattern:
x0 = 1
x1 = 5
x2 = 25
x3 = 125
x4 = 625
It seems that each term in the sequence is a power of 5. Furthermore, the exponent for each term appears to follow the pattern: the exponent is one less than the index of the term.
Step 3: Derive the general formula
Based on the pattern identified, we can formulate the general formula for the sequence as follows:
xk = 5^(k-1)
Therefore, the general formula for xk is xk = 5^(k-1).
Note: This formula assumes that k starts from 1. If k starts from 0, the formula would be xk = 5^k.
does anyone have lesson 10 linear function test answers
You have to be more precise in your notation, you probably meant:
x k+2 = 3x k+1 + 10x k
that is:
x 0 = 1 = 5^0
x 1 = 5 = 5^1
x 2 = 3(5) + 10(1) = 25 = 25(1) = 5^2
x 3 = 3(25) + 10(5) = 125 = 25(5) = 5^3
x 4 = 3(125) + 10(25) = 625 = 25(25) = 5^4
x5 = 3(625) + 10(125) = 3125 = 25(125) = 5^5
..... = 3(3125) + 10(625) = 15625 = 5^6
looks like x k = 5^k