uppose that the sequence x0, x1, x2... is defined by x0 = 4, x1 = 2, and xk+2 = 4xk+1+5xk for k≥0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. .
To find a general formula for the sequence xk, we can start by examining the first few terms and observe any patterns.
Given:
x0 = 4
x1 = 2
xk+2 = 4xk+1 + 5xk
Let's calculate the next few terms to look for a pattern:
x2 = 4x1 + 5x0 = 4(2) + 5(4) = 23
x3 = 4x2 + 5x1 = 4(23) + 5(2) = 102
x4 = 4x3 + 5x2 = 4(102) + 5(23) = 461
From the calculations above, we can observe the following pattern:
x0 = 4
x1 = 2
x2 = 23
x3 = 102
x4 = 461
To find a general formula, let's look at the differences between consecutive terms:
x1 - x0 = 2 - 4 = -2
x2 - x1 = 23 - 2 = 21
x3 - x2 = 102 - 23 = 79
x4 - x3 = 461 - 102 = 359
We can see that the differences between consecutive terms are not constant. However, let's take the differences between consecutive differences:
(21 - (-2)) = 21 + 2 = 23
(79 - 21) = 79 - 21 = 58
(359 - 79) = 359 - 79 = 280
From this, we can observe that the differences between consecutive differences are constant. This suggests that we may have a quadratic relationship.
Let's assume that the general formula for xk is of the form xk = ak^2 + bk + c, where a, b, and c are constants.
To find the constants, we substitute the initial values into the assumed general formula:
x0 = a(0)^2 + b(0) + c = 4
x1 = a(1)^2 + b(1) + c = 2
From these two equations, we can set up a system of equations to solve for a, b, and c.
Equation 1: a(0)^2 + b(0) + c = 4
Simplifies to: c = 4
Equation 2: a(1)^2 + b(1) + c = 2
Simplifies to: a + b + c = 2
Since c = 4, we have: a + b + 4 = 2
Rewrite the equation: a + b = -2
Now, we have two equations:
a + b = -2
c = 4
From the second equation, we have c = 4.
Substituting c into the first equation, we have a + b = -2.
Let's assume b = t, where t is a constant.
Then, a + t = -2.
Solving for a, we have a = -2 - t.
Assuming t = -1, we have a = -2 - (-1) = -1.
Therefore, the general formula for the sequence xk is:
xk = -k^2 - k + 4
To find a general formula for the sequence xk, we can start by examining the first few terms and look for any patterns.
Given that x0 = 4 and x1 = 2, we can calculate the next few terms:
x2 = 4x1 + 5x0 = 4(2) + 5(4) = 8 + 20 = 28
x3 = 4x2 + 5x1 = 4(28) + 5(2) = 112 + 10 = 122
x4 = 4x3 + 5x2 = 4(122) + 5(28) = 488 + 140 = 628
By calculating a few more terms, we can observe that the sequence follows a linear recurrence relation of the form xk+2 = axk+1 + bxk, where a = 4 and b = 5.
To find a general formula, we first assume that the formula is of the form xk = r^k. By substituting this assumption into the recurrence relation, we get:
r^k+2 = 4r^k+1 + 5r^k
Dividing both sides by r^k, we get:
r^2 = 4r + 5
This is a quadratic equation, which can be rearranged as:
r^2 - 4r - 5 = 0
Factoring the quadratic equation, we have:
(r - 5)(r + 1) = 0
This gives us two possible values for r: r = 5 and r = -1.
Now, we can write the general formula for xk as a linear combination of these two values:
xk = Ar^k + B(-1)^k
where A and B are constants that need to be determined.
To find the values of A and B, we use the initial conditions x0 = 4 and x1 = 2:
For x0: 4 = A(5^0) + B(-1)^0 = A + B
For x1: 2 = A(5^1) + B(-1)^1 = 5A - B
Solving these two equations simultaneously, we find:
A = 6/7 and B = 22/7
Therefore, the general formula for xk is:
xk = (6/7)(5^k) + (22/7)(-1)^k