Find and solve a recurrence equation for the number g_n of ternary strings of length n that do not contain 102 as a substring. g_n counts the number of strings of length n so I know g_1=3 g_2=9 and g_3=26. I think g_n=3g_(n-1)-g_(n-3)?
Answers
To prove this relation, let's consider the last digit of the string of length n.
It can be either 0, 1, or 2. If it is 0 or 1, we can append either 0, 1, or 2 to it to get a valid string of length n+1. Thus, there are 3 * g_{n-1} valid strings of length n+1 that end in 0 or 1.
If the last digit is 2, then the second-last digit cannot be 1 (otherwise we have the substring 102). So the second-last digit can be either 0 or 2. If it is 0, we can append any of the three digits to get a valid string of length n+1. If it is 2, we can only append 0 or 1 (since appending 2 would create the substring 102).
Thus, there are 2 * g_{n-2} valid strings of length n+1 that end in 2.
Therefore, we have:
g_{n+1} = 3 * g_{n-1} + 2 * g_{n-2}
To solve this recurrence relation, we can use characteristic equations. Assume that g_n = r^n for some constant r. Substituting this into the recurrence relation, we get:
r^{n+1} = 3r^{n-1} + 2r^{n-2}
Dividing both sides by r^{n-2} (which is nonzero since r ≠ 0), we get:
r^3 - 3r + 2 = 0
This is the characteristic equation. We can factor it as:
(r-1)^2 (r+2) = 0
Thus, the roots are r=1 (with multiplicity 2) and r=-2. Therefore, the general solution to the recurrence relation is:
g_n = c_1 + c_2 * n + (-2)^n * c_3
where c_1, c_2, and c_3 are constants determined by the initial conditions. Using the initial values g_1=3, g_2=9, and g_3=26, we get the following system of equations:
c_1 - 2c_3 = 3
c_1 + c_2 - 4c_3 = 9
c_1 + 2c_2 + 4c_3 = 26
Solving this system, we get c_1 = 2, c_2 = 1, and c_3 = -1/2. Thus, the solution to the recurrence relation is:
g_n = 2 + n - 2(-2)^n / 2
or equivalently,
g_n = n + 2 - (-2)^n
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Related Questions
1. construct a 5-to-32 line decoder with four 3-to-8 line decoder with enable and a 2-to-4 line decoder. use block diagrams for the components, label all inputs and outputs.
Answers
Sure! Here's a block diagram representation of a 5-to-32 line decoder using four 3-to-8 line decoders with enable (3:8 decoder with enable) and a 2-to-4 line decoder.
_________ _________ _________ _________
| | | | | | | |
IN[4] | 5:32 | | 3:8 | | 3:8 | | 3:8 |
----->| Decoder |---->| Decoder |---->| Decoder |---->| Decoder |----> Y[31]
|_________| |_________| |_________| |_________|
| | | |
IN[3] | | | |
----->| | | |
| 3:8 Decoder | | 3:8 Decoder |
|___________________________| |___________________________|
| | | |
IN[2] | | | |
----->| | | |
| 3:8 Decoder | | 3:8 Decoder |
|___________________________| |___________________________|
| | | |
IN[1] | | | |
----->| | | |
| 3:8 Decoder | | 3:8 Decoder |
|___________________________| |___________________________|
| | | |
IN[0] | | | |
----->| 2:4 Decoder | | |
|___________________________| | 3:8 Decoder |
|___________________________|
Inputs:
IN[4:0]: 5-bit input lines
Outputs:
Y[31:0]: 32 output lines
The 5-to-32 line decoder takes a 5-bit input (IN[4:0]) and produces 32 output lines (Y[31:0]). It uses four 3-to-8 line decoders with enable (3:8 Decoder) to decode the input bits and generate intermediate outputs. The intermediate outputs are then connected to a 2-to-4 line decoder (2:4 Decoder) to produce the final 32 output lines (Y[31:0]).
Note: The enable lines for the 3-to-8 line decoders are not shown in the diagram for simplicity. Each 3-to-8 line decoder will have its own enable input, which can be used to enable or disable the decoder's functionality.
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The value of an independent variable from the prior period is referred to as...Question 15 options:lagged variable.dummy variable.predictor variable.categorical variable.
Answers
The value of an independent variable from the prior period is referred to as a lagged variable. Therefore, the correct option is A.
This is because the value is based on data from a previous time period and is used to predict the current value of the dependent variable. A lagged variable is important in time series analysis, where the values of a variable over time are analyzed to identify trends and patterns.
By including lagged variables in a model, we can account for the influence of past values on the current value, which can help to improve the accuracy of our predictions. Therefore, when working with time series data, it is important to include lagged variables as predictors in our models. Hence, the correct answer is option A.
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$2,300 deposit earning 3.9% compounded after 2 years , what will the balance be after 2 years
Answers
The balance after 2 years for a $2,300 deposit earning 3.9% compounded annually will be $2,552.72. To calculate the future balance of a deposit after a certain period of time, we can use the compound interest formula:
Future Balance = Principal × (1 + Interest Rate/100)^Number of Years
In this case, the principal is $2,300, the interest rate is 3.9%, and the number of years is 2. Plugging these values into the formula, we get:
Future Balance = $2,300 × (1 + 3.9/100)^2
First, we'll find the value inside the parentheses:
(1 + 3.9/100) = (1 + 0.039) = 1.039
Now, raise this value to the power of 2:
1.039^2 = 1.079521
Next, multiply the principal by the result:
$2,300 × 1.079521 = $2,552.72
So, the balance after 2 years for a $2,300 deposit earning 3.9% compounded annually will be $2,552.72.
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Find the value of a in the equation below.
5 = x - 18
Answers
Answer:
There's no A so I'm going to assume you meant X
X = 23
Step-by-step explanation:
X is equal to 23, because 23 - 18 = 5
or 5 + 18 = 23
From the equation, find the axis of symmetry of the parabola. f(x) = 10x2 - 20x + 16
a. x = -6
b. x = 6
c. x = 1
d. x = -1
PLEASE HURRY!
Answers
Step-by-step explanation:
The answer is C!!! to your question.
A family pass to the amusement park costs $54. Using Distributive Property, write an expression that can be used to find the cost in dollars of 8 family passes. (I'm actu
Answers
The cost of 8 family passes to the amusement park is $432 when a family pass to the amusement park costs $54.
To find the cost of 8 family passes to the amusement park, we can use the Distributive Property of multiplication over addition. The Distributive Property states that a(b + c) = ab + ac, where a, b, and c are numbers. In this case, we can use the Distributive Property to write:
8 x $54 = (5 x $54) + (3 x $54)
We can see that we have broken 8 down into two smaller numbers that we can multiply by $54 individually, and then add together. The first term, 5 x $54, represents the cost of 5 family passes, while the second term, 3 x $54, represents the cost of 3 family passes. Adding these two values together, we get:
(5 x $54) + (3 x $54) = $270 + $162 = $432
Therefore, the cost of 8 family passes to the amusement park is $432.
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can someone pls help with this question
Answers
Answer:
26
Step-by-step explanation:
Use the Pythagorean Theorem: \(a^2+b^2=c^2\)
what is 3/4 divide by 1/6
no simplfly
Answers
Answer:
18/4
After simplification
9/2
Step-by-step explanation:
3/4 ÷ 1/6
Copy dot flip
3/4 * 6/1
18/4
If we simplify
Divide the top and bottom by 2
9/2
latifa writes down a number. she says
if i find a cube root of that number and then quadruple the answer, i get 16
what number did latifa write down?
Answers
Answer:
8
Step-by-step explanation:
the cubed root of 8 equals 2
if you quadruple 2 (multiply it by 4), you will get 8
other recipe calls for 34
3
4
cup of milk. Marisa has 1
1
cup of milk.
Does Marisa have enough milk for both recipes?
Answers
Which graph represents the solution set of the compound inequality?
−5
please help!!!
Answers
Answer:
the first
Step-by-step explanation:
it just is. Be smarter.
Must answer in integers only and reduced fractions if needed and no decimals
Answers
ANSWER
\(x=3\)EXPLANATION
We want to solve the equation using a common base. Since the bases on both sides of the equation are equal, the exponents are also equal.
This implies that:
\(3x-7=5-x\)Collect like terms and simplify:
\(\begin{gathered} 3x+x=5+7 \\ 4x=12 \\ x=\frac{12}{4} \\ x=3 \end{gathered}\)That is the solution.
What happens when a triangle is dilated using one of the vertices as the center of the dilation? Complete
the explanation.
The sides of the triangles adjacent to the center of dilation will be collinear. The third sides of the pre-
image and image will be ?
The vertex used as the center of dilation will be in
Answers
Answer:
The sides of the triangles adjacent to the center of dilation will be collinear. The third sides of the pre-
image and image will be “parallel”
The vertex used as the center of dilation will be in “the same location in both triangles”
Ana is solving the following system of equations using elimination by addition. What is the new equation after eliminating the x-terms?
2x+3y=4
−2x+5y=60
Answers
The new equation after eliminating the x-terms is 8y = 64
How to determine the new equation after eliminating the x-terms?From the question, we have the following parameters that can be used in our computation:
2x+3y=4
−2x+5y=60
Express properly
So, we have
2x + 3y = 4
−2x + 5y = 60
Add the two equations to eliminate x
So, we have
3y + 5y = 4 + 60
Evaluate the like terms
8y = 64
Hence, the new equation after eliminating the x-terms is 8y = 64
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Which equation represents “the product of a number and 24 is 72”? 24 p = 72 24 = 72 p (24) (72) = p StartFraction p Over 24 EndFraction = 72
Answers
Answer:
24p = 72
Step-by-step explanation:
Let the number be p.
The product of p and 24 equals to 72.
p × 24 = 72
24p = 72
Answer:
24p = 72
Step-by-step explanation:
A
if each option is equally likely within each category what is the probability that the lego figure does not have blue legs but has a red body?
Answers
The probability that the LEGO figure does not have blue legs but has a red body is 2/9. so, the correct answer is B).
To find the probability that the LEGO figure does not have blue legs but has a red body, we need to first find the probability of the figure having a red body, which is 1/3.
Next, we need to find the probability of the figure not having blue legs. Since there are three options for legs, and one of them is blue, the probability of the figure having blue legs is 1/3. Therefore, the probability of the figure not having blue legs is 1-1/3, which simplifies to 2/3.
Finally, we multiply the probabilities of the two events to get the probability that the figure has a red body and does not have blue legs: (1/3) x (2/3) = 2/9.
Therefore, the answer is (B) 2/9.
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--The given question is incomplete, the complete question is given
" A LEGO action figure is created for visitors that tour a factory. Each figure consists of a randomly chosen set of legs, body, head, and hair/hat. The following table shows the available options for each category.
If each option is equally likely within each category what is the probability that the LEGO figure does NOT have blue legs but has a red body.
1/6
2/9
1/10
1/3"--
Help me solve this please
Answers
Answer:
0
Step-by-step explanation:
3. Growth of Cholera bacteria. Suppose that the cholera bacteria in a colony grows unchecked according to the Law of Exponential Change. The colony starts with 1 bacterium and doubles in number every half hour. a. How many bacteria will the colony contain at the end of 24 h
Answers
Answer:
2.814 * 10^14
Step-by-step explanation:
For exponential growth:
A = A0*e^kt
A = final amount ; A0 = initial amount ; t = time
Since bacterium doubles ever half hour ;
In 1 hour number of bacterium will be 2² = 4
Hence
Final amount after 1 hour
4 = 1*e^k*1
4 =e^k
Take In of both sides
In(4) = k
Number of bacterium in 24 hours
A = 1*e^In4 * 24
A = e^24In4
A = e^33.271064
A = 281474976710656
A = 2.814 * 10^14
The radius of the small circle is 4cm and the area of the large circle is 67.12 square centimeters. calculate the area of the shaded region.
Answers
Answer:
Did you multiply them by 3.14?
Step-by-step explanation:
Answer:
the area of the shaded region is 16.88
Step-by-step explanation:
A:π(4)² A of large circle=67.12
A=50.24 67.12-50.24=16.88
the area of the shaded region is 16.88
sorry if it's not right, i tried.
What are the measurements rounded to the nearest hundredth.
Answers
Answer:
Step-by-step explanation:
The numbers 1 through 8 are to be placed in the circles in the diagram shown such that the sum of the numbers on each side of the figure is 15. Find the values of x, y, and z. Then copy the diagram on a piece of paper and fill in each circle with the appropriate number. Be sure to show all your work. Upload the finished problem below.
Note: x satisfies –4(x – 2) = –4,
y satisfies 9 + 2(y – 3) = 3(y – 2) + 1,
and z satisfies z/3 + 1/2 = 5/2.
Answers
Answer:
ff
Step-by-step explanation:
f
A scale drawing of an office space uses a scale of 1 in : 5 yd. The scale drawing has an area of 12 square inches. What is the area of the actual office space
Answers
Answer:
60 Yards.
Step-by-step explanation:
Ooooooooo Proportion question!
We can write an equivalent fraction here:
\(\frac{1}{5} =\frac{12}{x}\)
Because 12 wold be equivalent to the 1 inches.
We can multiply by 12 because 1*12 would be 12.
5*12=60.
x=60, which is the answer.
3x times 4=7x-(x+17) please help me
Answers
Answer:
x = ⁻¹⁷⁄₆ or -2.83333333333Explanation:
3x · 4 = 7x - (x + 17)
Simplify both sides of the equation
3x (4) = 7x − (x + 17)
3x (4) = 7x + −1 (x + 17) (Distribute the Negative Sign)
3x (4) = 7x + −1x + (−1) (17)
3x (4) = 7x + −x + −17
12x = 7x + −x + −17
12x = (7x + −x) + (−17) (Combine Like Terms)
12x = 6x + −17
12x = 6x − 17
Subtract 6x from both sides
12x − 6x = 6x − 17 − 6x
6x = −17
Divide both sides by 6
6x / 6 = −17 / 6
Simplify
x = ⁻¹⁷⁄₆
Turn the fraction into a decimal
-17 ÷ 6 = -2.83333333333
x = ⁻¹⁷⁄₆ or -2.83333333333Which of the following expressions is incorrect? A. |-12| = 12 B. |12| = 12 C. -|-12| = -12 D. -|12| = 12
Answers
Answer:
D
Step-by-step explanation:
Answer:
A and D
Step-by-step explanation:
A and D
prove that any graph of minimum degree at least three contains a cycle of even length.
Answers
Answer:
a cycle is a sequence of non-repeated vertices and the degree of a graph is the number of neighbors the vertex has.
Six consecutive odd numbers sum up to
36, find the numbers
Answers
Answer:
1+3+5+7+9+11
Step-by-step explanation:
Anita paid a total of $57.50 for a blazer. The price of the blazer was $52.85. What was the sales tax rate?
Answers
Answer:
8.79 % tax rate
Step-by-step explanation:
total paid = tax + price
57.50 = tax +52.85
tax = 4.65
tax rate =( tax ÷ price before the tax) × 100%
= (4.65÷ 52.85) × 100%
= 8.79%
Answer:8.08%
Can you solve?
|x+5|=12
What is the answer please
Answers
x + 5 = 12
x = 7
X + 5 = -12
x = -17
The answer is 7 and -17
what is the distribution of the difference between sample means from two normal populations?
Answers
The distribution of the difference between sample means from two normal populations is also normal.
Specifically, if we have two independent random samples of sizes n1 and n2, drawn respectively from two normal populations with means μ1 and μ2 and standard deviations σ1 and σ2, the difference between the sample means (x1 - x2) follows a normal distribution with mean μ1 - μ2 and standard deviation given by:
σ(x1 - x2) = sqrt((σ1^2 / n1) + (σ2^2 / n2))
This formula is derived from the fact that the variance of the difference between two random variables is the sum of their variances if the two variables are independent. In this case, the random variables are the sample means x1 and x2.
It is important to note that this result holds only if the two populations are normal and the samples are independent. If the populations are not normal, but the sample sizes are large (i.e., n1 and n2 are both greater than or equal to 30), the Central Limit Theorem can be used to approximate the distribution of the difference between sample means as normal.
The normal distribution of the difference between sample means has important applications in statistical inference, such as in the construction of confidence intervals and hypothesis testing. For example, we can use this distribution to test whether two population means are equal by comparing the difference between their sample means to a hypothesized value of zero, using the formula for the test statistic:
t = (x1 - x2 - d) / sqrt((s1^2 / n1) + (s2^2 / n2))
where d is the hypothesized difference between the population means and s1 and s2 are the sample standard deviations. This test statistic follows a t-distribution with degrees of freedom equal to (n1 - 1) + (n2 - 1), assuming equal variances in the two populations. If the p-value associated with this test statistic is less than the chosen level of significance, we reject the null hypothesis that the two population means are equal.
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.The figure below includes a semicircle and
trapezoid. Find the area of the figure.
Answers
Answer:
244.93
Step-by-step explanation:
area formula for trapezoid:1/2*(b1+b2)*h
area formula for semi circles: (3.14*r^2)/2
so then we cut the figure to make a semi cirlce and trapazoid
semicircle- 14 is the full diameter, divide by 2 to make 7, 7×3.14×7)/2=76.93
trapezoid: base one is 18. base 2 is 14. 18+14=32 then times that by 10.5 which is 336 and divide that by 2 is 168
add both and you get 244.93