An equation that helped to find the value of x is 30 = 4x.
The value of x is 7.5mm.
What is the perimeter?A closed shape's perimeter is the sum of the lengths of its outside boundaries. The lengths of all the sides are added to determine the measurement.
Given:
A shape is a square.
And perimeter of the square is 30 mm.
And we know that the sides of the square are equal.
So, the perimeter of the square = 4 x The length of one side.
Substituting all the given values,
we get,
30 = 4x
x = 30/4
x = 7.5 mm
Therefore, x = 7.5 mm.
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The population of Lcavetown is 123,000 people, and is decreasing at a rate of 9,4% per year What will the population of Leavetown be 100 years from now? (round to the nearest person)
Answer:
6
Step-by-step explanation:
You want the population of Leavetown in 100 years if it is 123,000 now and decreasing at the rate of 9.4% per year.
Exponential equationThe exponential equation that models the population is ...
p = 1230000·(1 -0.094)^t
where t is the number of years later.
ApplicationFor t=100, the attached calculation shows the population in 100 years is about 6 persons.
In 2000, a forest covered an area of 1500 km². Since then, this area has decreased by 6.25% each year.
Lett be the number of years since 2000. Let y be the area that the forest covers in km².
Write an exponential function showing the relationship between y and t.
The relationship between y and t can be modeled by an exponential function of the form:
y = a x e^(-rt)
What are exponential functions?An exponential function is a mathematical function which we write as a
f(x) = aˣ, where a is constant and x is variable term. The most commonly used exponential function is eˣ , where e is constant having value 2.7182
The relationship between y and t can be modeled by an exponential function of the form:
y = a x e^(-rt)
where a is the initial area of the forest (1500 km²), r is the rate of decrease (6.25%), and t is the number of years since 2000.
To find the value of r, we can convert 6.25% to a decimal:
r = 0.0625
Now we can plug in the values for a and r into our exponential function:
y = 1500 x e^(-0.0625t)
This exponential function shows the relationship between the area of the forest and the number of years since 2000.
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1. Jonathan's family had a pizza party with their neighbors, and they ordered 7 pizzas.
Everyone ate 1 1/4 pepperoni pizza, 2 3/4 sausage pizza, and 3/4 of the cheese pizza. How
much pizza was leftover after the party?
Answer: 2.25 slices which is 9/4 slices i think im right
Step-by-step explanation:
add 1 1/4+2 3/4+3/4 which makes 4.75 slices or 19/4 slices do 7-4.75=2.25/9/4 slices
Predict the missing component in the nuclear equation.
The missing component in the nuclear equation.137Cs55 → X + -e will be 137Ba56. option B is correct.
What is nuclear reaction?The particles in the nucleus are changed, and one element is transformed into another element when particles in the nucleus are gained or lost.
Nuclear reactions are processes in which one or more nuclides are produced from the collisions between two atomic nuclei or one atomic nucleus and a subatomic particle.
This was the first observation of an induced nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus.
The most notable man-controlled nuclear reaction is the fission reaction which occurs in nuclear reactors. A target nucleus is within the range of nuclear forces for the time allowing for numerous interactions between nucleons.
therefore, option B is correct.
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A game manufacturer is designing a pocket travel-edition of one of its top-selling games. The tull-size triangular game board and the proposed travel board are represented below
where ABC-A XYZ and the given side lengths are measured in centimeters
32
52
Which of these are true? Choose all that are comect
The perimeter of the full-size game board is six times the perimeter of the travel-edition game board
The travel-edition side comesponding to the 48-cm side on the tull-size board will measure 13 centimeters.
The travel-edition side comesponding to the 48-cm side on the full-size board will measure 12 centimeters
The three angles represented on the travel-edition board are congruent to their comesponding angles on the full-size board
Based on the given information and measurements, the following statements are true:
What are the true statements?The perimeter of the full-size game board is six times the perimeter of the travel-edition game board: This statement is false. The perimeter of the full-size game board can be calculated as the sum of the lengths of its three sides: AB + BC + AC = 32 + 52 + 52 = 136 cm. The perimeter of the travel-edition game board can be calculated as the sum of the lengths of its three sides: AX + XY + YZ = 13 + 21 + 18 = 52 cm. Therefore, the perimeter of the full-size game board is 2.62 times the perimeter of the travel-edition game board, not 6 times.
The travel-edition side corresponding to the 48-cm side on the full-size board will measure 12 centimeters: This statement is false. The travel-edition side corresponding to the 48-cm side on the full-size board is side XY. The length of XY can be calculated using proportions: XY/48 = YZ/52, which gives XY = 48*18/52 = 16.62 cm (rounded to two decimal places).
The travel-edition side corresponding to the 48-cm side on the travel-edition board will measure 13 centimeters: This statement is true. The travel-edition side corresponding to the 48-cm side on the travel-edition board is side AX, which has a length of 13 cm.
The three angles represented on the travel-edition board are congruent to their corresponding angles on the full-size board: This statement is true. The three angles of a triangle are determined by the lengths of its sides, so if two triangles have the same side lengths, their angles must be congruent as well.
Therefore, the three angles represented on the travel-edition board (at vertices A, X, and Y) are congruent to their corresponding angles on the full-size board (at vertices A, B, and C).
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Richard used a radius measure of a circle to be 3.6 inches when he calculated the area of a circle. The correct radius measure was actually 3.5 inches. What is the difference between Richard’s measured area of the circle and the actual area of the circle?
a. 0.1π square inches
b. 0.71π square inches
c. π square inches
d. 0 square inches
e. 2.4 square inches
As a result, the answer is (e) 2.4 square inches, which is the difference between Richard's measured and real circle area.
What is area?In mathematics, area is a measure of the amount of space occupied by a two-dimensional object, such as a rectangle, triangle, circle, or any other shape. It's a scalar quantity that describes the size of a region in two-dimensional space. The units of area are typically square units, such as square inches, square centimeters, square meters, etc. In general, the area of a shape is a measure of how much space it occupies, and it is an important concept in geometry, engineering, and many other fields.
Here,
The formula for the area of a circle is given by:
A = πr²
Where r is the radius of the circle.
Using the incorrect radius of 3.6 inches, the calculated area would be:
A = π * (3.6 inches)² = 40.44 square inches
Using the correct radius of 3.5 inches, the actual area would be:
A = π * (3.5 inches)² = 38.5 square inches
So, the difference between Richard's measured area of the circle and the actual area of the circle would be:
40.44 square inches - 38.5 square inches = 1.94 square inches
Therefore, the answer is (e) 2.4 square inches that is the difference between Richard’s measured area of the circle and the actual area of the circle.
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Bacteria colonies can increase by 45% every 7 days. If you start with 200 bacteria microorganisms, how large would the colony be after 35 days? First, identify I, the starting amount. Future Amount = [?](1+ Remember: Future Amount = (1 + r)t
After 35 days, the colony will be 1281.95.
What are Exponential Functions?Exponential functions are functions where the independent variable, x is in the exponent.
Given that,
Bacteria colonies can increase by 45% every 7 days.
We know that,
Future amount = l (1 + r)^t
where l is the starting amount, r is the growth rate and t is the time.
Here,
l = 200
r = 45% = 45 / 100 = 0.45
t = 35 / 7 = 5
Substituting,
Future amount = (200) (1 + 0.45)⁵
= 1281.95
Hence the required amount is 1281.95.
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Create a scatter plot with the data. What is the correlation of this scatter
plot?
The solution is given below.
What is scatter plot?A scatter plot is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. If the points are coded, one additional variable can be displayed.
here, we have,
A scatter plot is a graph in which the values of two variables are plotted along two axes. Every data of the table is the coordinate of a point in the plot. To make a scatter plot, first assign a variable to x-axis and the other variable to y-axis, in this question speed was assigned to x-axis and distance was assigned to y-axis. And then, locate the points as coordinates, for example, the first point is (2, 5).
See picture attached.
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at 12/31 accounting records of Gordon Inc contain
AP $2500, Building $31250, Land $30,000, Notes Payable $ ?, Retained earnings
$125,000, AR $18750, Cash ?, Equipment $40,000, Captial Stock $12500
If the cash balance at 12/31 is $67,500 The notes payable balance is
To find the Notes Payable balance, we need to use the accounting equation:
Assets = Liabilities + Equity
We are given the following information:
AP = $2,500
Building = $31,250
Land = $30,000
Notes Payable = ?
Retained Earnings = $125,000
AR = $18,750
Cash = $67,500
Equipment = $40,000
Capital Stock = $12,500
We can add up the assets to get:
Assets = AP + Building + Land + AR + Cash + Equipment
= $2,500 + $31,250 + $30,000 + $18,750 + $67,500 + $40,000
= $190,000
We can rearrange the accounting equation to solve for liabilities:
Liabilities = Assets - Equity
Plugging in the values we have:
Liabilities = $190,000 - ($125,000 + $12,500)
= $190,000 - $137,500
= $52,500
Therefore, the Notes Payable balance is:
Notes Payable = Liabilities - AP
= $52,500 - $2,500
= $50,000
7. a certain college graduate borrows $8000 to buy a car. the lender charges interest at an annual rate of 10%. assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k, determine the payment rate k that is required to pay off the loan in 3 years. a
The payment rate that is required to pay off the loan in 3 years is $3139.88 per year. The borrower will pay approximately $1695.64 in interest over the 3-year period.
The differential equation that models this situation is:
dy/dt = 0.10y - k
where y(t) is the amount owed at time t, 0.10 is the annual interest rate, and k is the annual payment rate. The initial condition is y(0) = $8000.
The first term on the right-hand side represents the interest that accumulates on the loan, and the second term represents the payments made by the borrower.
To determine the payment rate k that is required to pay off the loan in 3 years, we need to solve the differential equation with the initial condition y(0) = $8000 and the terminal condition y(3) = 0.
The general solution to the differential equation is:
y(t) = (8000/k) e^(0.10t) - (8000/k)
Setting t = 3 and y(3) = 0, we get:
0 = (8000/k) e^(0.30) - (8000/k)
Solving for k, we get:
k = 3139.88
Therefore, the payment rate that is required to pay off the loan in 3 years is $3139.88 per year.
To determine how much interest is paid during the 3-year period, we can integrate the interest rate over the time interval [0, 3]:
∫[0,3] 0.10y(t) dt = ∫[0,3] 0.10[(8000/k) e^(0.10t) - (8000/k)] dt
= (8000/k) [e^(0.30) - 1] - 2400
Substituting k = 3139.88, we get:
∫[0,3] 0.10y(t) dt ≈ $1695.64
Therefore, the borrower will pay approximately $1695.64 in interest over the 3-year period.
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____The given question is incomplete, the complete question is given below:
A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. Assume that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k. (a) Write a differential equation that models this situation, including the initial condi- tion. (b) Determine the payment rate k that is required to pay off the loan in 3 years. (c) Determine how much interest is paid during the 3-year period.
Find each angle measure to the nearest degree.
3) sin U = 0.9945
4) tan C = 3.7321
The measure of angle U = 84° and the measure of angle C = 75°
The inverse of a trigonometric function.The sine, cosine, tangent, cotangent, secant, and cosecant functions are the fundamental trigonometric functions. Inverse trigonometric functions are just the inverse functions of these functions. They can also be referred to as antitrigonometric functions or arcus functions. To find the angle for any trigonometric ratio, these inverse trigonometric functions are applied.
Given that:
sin U = 0.9945
U = sin⁻¹ (0.9945)
U = 83.988°
U ≅ 84° ( to the nearest degree)
tan C = 3.7321
C = tan⁻¹ (3.7321)
C = 75°
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d. In this situation, what does the solution to the equation C(t) = 2 tell us? Find
that solution.
e. Write an equation that would allow us to find the age of the car when we know
C(t).
ora 1 Unit 4
n 17
CC BY 2019 by Illustrative Mathematics®
The equation C(t) = 2 tells us that at a moment of t, the numeric value of the variable C is of 2 units.
How to define the ordered pair and how it relates to the numeric value?The general format of an ordered pair is given as follows:
(x,y).
The meaning of an ordered pair (x,y) is that y = f(x), meaning that the numeric value of the function at the value of x is of y.
The ordered pair for this problem is given as follows:
(t, C(t)) = (t,2).
Meaning that at a moment of t, the numeric value of the variable C is of 2 units.
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(book exercise 1.2.37 determine in terms of h the first two terms and the error term in the taylor series for ln(3-2h)
Answer:
which book are u talking about.. post the question please..
Decide how many solutions this equation has:
x2 - 2x + 1 = 0
A customer bought an item for N$640 and paid N$160 down with an agreement to pay the balance plus a charge fee of N$16 in three months. Find the simple interest rate at which the customer was paying for the item.
The simple interest rate for the item is 13 1/3%.
What is the simple interest rate?
Simple interest is the charge on borrowing calculated as a linear function of the amount borrowed, time and the interest rate.
Interest rate = interest / (time x amount borrowed)
Interest = N$16time = 3/12 = 0.25 Amount borrowed = N$640 - N$160 = N$480Interest rate = N$16 / (N$480 x 0.25)
= 0.13333 = 13 1/3 %
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a monkey presses the keys on a typewriter randomly. suppose the typewriter only contains the 26 letters of the alphabet (no space or other characters), and the monkey chooses the next key to press uniformly at random from the 26 options. suppose the monkey types n letters in total.
The probability of typing any particular sequence of n letters is (1/26)^n. The probability of word of length m (where m ≤ n) is (1/26)^m. The probability of at least one occurrence is 1 - (25/26)^(n-m+1). The number of times the monkey types a specific word of length m is (n-m+1)*(1/26)^m.
Assuming that the monkey's key presses are truly random and independent of each other, the probability of the monkey typing any particular sequence of n letters is (1/26)^n. This is because there are 26 choices for each letter, and the probability of the monkey choosing any particular letter is 1/26.
The probability of the monkey typing a specific word of length m (where m ≤ n) is (1/26)^m. This is because the monkey must type the specific sequence of m letters in order, and each letter has a probability of 1/26 of being typed.
The probability of the monkey typing at least one occurrence of a specific word of length m is 1 - (25/26)^(n-m+1). This is because the monkey has n-m+1 opportunities to type the word, and the probability of missing the word in any one opportunity is 25/26 (since there are 25 other letters that the monkey could type instead).
The expected number of times the monkey types a specific word of length m is (n-m+1)*(1/26)^m. This is the product of the probability of the monkey typing the word in any given opportunity (which is (1/26)^m), and the number of opportunities the monkey has to type the word (which is n-m+1).
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Contessa is solving an absolute value inequality. She writes the compound inequality -7<=8-3q<=7 as her first step. Which inequality is Contessa solving?
The inequality that Contessa is solving is obtained as 5 ≥ q ≥ 1/3.
What is an inequality?
In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.
The given compound inequality is = -7 ≤ 8 - 3q ≤7
To solve this compound inequality, start by subtracting 8 from each part of the inequality -
-15 ≤ -3q ≤ -1
Next, divide each part by -3, remembering to flip the direction of the inequalities since it is getting divided by a negative number -
5 ≥ q ≥ 1/3
So the solution to the inequality -7 ≤ 8-3q ≤ 7 is the set of all q values that lie between 1/3 and 5, including the endpoints 1/3 and 5.
Note that this solution set is different from the solution set of the absolute value inequality |8-3q| < 7, which does not include the endpoints.
Therefore, the inequality is 5 ≥ q ≥ 1/3.
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Add. Write your answer in simplest form.
3/8 + 2/5 WILL GIVE BRAINLIST IF RIGHT!!!
If angle C and angle E are opposite angles of parallelogram CDEF, then they are supplementary.
true or false
Answer:
False
Step-by-step explanation:
This is because the opposite angles would be congruent, or equal.
I hope this helps! :)
Consider the following parametric equation.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x=−10cos2t, y=−10sin2t; 0≤t≤π
a. Eliminating the parameter yields the equation [tex]x2+y2=100[/tex], which is a circle centered at the origin with a radius of 10.
b. The curve is a circle with a positive orientation, going counterclockwise from the origin.
a. To eliminate the parameter, we first square both sides of the equations to obtain: [tex]x2=(−10cos2t)2 and y2=(−10sin2t)2.[/tex]Then, since cos2t and sin2t are both between -1 and 1, the terms on the right hand side of each equation can be simplified to 100. Thus, the equation [tex]x2+y2=100[/tex]is obtained.
b. This equation describes a circle centered at the origin with a radius of 10. The positive orientation of the curve is counterclockwise from the origin, i.e. it starts at the origin and moves up, then to the right, then down, and then to the left.
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Ella participated in a race each week for four weeks. She recorded her race times in this chart. What inequality about her race times is true?
Determine each product. a. (x-2) (3x+5)
Answer:
3x^2 - x - 10
Step-by-step explanation:
Foil (First, outside, inside, last)
3x^2 + 5x + -6x - 10
combine like terms
3x^2 - x - 10
find all numbers whose absolute value is -4
All absolute values are greater than or equal to zero so there is no such absolute value of - 4.
What is an absolute value function?We know the absolute value function of the modulus function always outputs a positive value irrespective of the sign of the input.
In piecewise terms | x | = x for x ≥ 0 and | x | = - x for x < 0.
We know, |a| = a, |- a| = a, and |0| is 0 therefore, The least possible value of a modulus is zero.
Therefore, There is no such numbers whose modulus value is - 4.
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Find an autonomous differential equation with all of the following properties:
equilibrium solutions at y=0 and y=3,
y' > 0 for 0 y' < 0 for -inf < y < 0 and 3 < y < inf
dy/dx =
The differential equation dy/dt = y(3-y) smug all of the given conditions.
One possible autonomous differential equation with equilibrium solutions at y=0 and y=3, and with y' > 0 for 0 < y < 3 and y' < 0 for -∞ < y < 0 and 3 < y < ∞, is:
dy/dt = y(3-y)
We can see that y=0 and y=3 are equilibrium solutions by setting dy/dt = 0 and solving for y:
dy/dt = y(3-y) = 0
y = 0 or y = 3
To check the sign of y', we can use the derivative of y(3-y) with respect to y: d/dy (y(3-y)) = 3 - 2y
For y < 0, we have y(3-y) < 0, so d/dy (y(3-y)) < 0, which says that y' < 0.
For 0 < y < 3, we have y(3-y) > 0, so d/dy (y(3-y)) > 0, which implies that y' > 0.
For y > 3, we have y(3-y) < 0, so d/dy (y(3-y)) < 0, which implies that y' < 0.
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Suppose the data have a bell-shaped distribution with a mean 30 and a standard deviation of 5. Use the empirical rule to determine the percentage of data between 15 and 45.
Select one:
a. 95.00%
b. 97.35%
c. 99.70%
d. 81.50%
According to the empirical rule, for a bell-shaped distribution with a mean of 30 and a standard deviation of 5 is approximately 99.7%,
Using the empirical rule, we know that for a normal distribution with a mean of 30 and a standard deviation of 5, approximately:
68% of the data falls within 1 standard deviation of the mean, which in this case is between 25 and 35.95% of the data falls within 2 standard deviations of the mean, which in this case is between 20 and 40.99.7% of the data falls within 3 standard deviations of the mean, which in this case is between 15 and 45.Therefore, the percentage of data between 15 and 45 is approximately 99.7%, which corresponds to option c.
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Estimate the mean of the number of homework problems completed by students in an hour given in the following grouped frequency table. • Round the final answer to one decimal place. (Please do not include the units in your answer.) Value Interval Frequency 3 3-6. 7-10 11-14 15-18 Provide your answer below: mean estimate QUESTION 21 . 1 POINT QUESTION 21 · 1 POINT A Trial best fits which of the following descriptions? Select the correct answer below: O a subset of the set of all outcomes of an experiment O one specific execution of an experiment O a planned activity carried out under controlled conditions a particular result of an experiment QUESTION 22.1 POINT If A and B are independent events, P(A) = 0.13, and P(B) = 0.72, what is P(BA)? Provide your answer below:
Let's denote the midpoint of each value interval by x and the frequency by f. Then the estimated mean is given by:
(mean) = (Σxf) / (Σf)
where the summation is taken over all value intervals.
Using the provided grouped frequency table, we can calculate the estimated mean as follows:
Midpoint (x) Frequency (f) xf
4.5 12 54
8.5 18 153
12.5 24 300
16.5 14 231
Total 68 738
The sum of the xf column is 738, and the sum of the f column is 68. Therefore, the estimated mean is:
(mean) = (Σxf) / (Σf) = 738 / 68 ≈ 10.85
Answer to Question 22:
Since A and B are independent events, we have:
P(B|A) = P(B)
Also, from the definition of conditional probability, we have:
P(B|A) = P(BA) / P(A)
Solving for P(BA), we get:
P(BA) = P(B) * P(A) = 0.13 * 0.72 = 0.0936
Therefore, P(BA) = 0.0936.
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Choose the number of possible combinations shown by this tree diagram.
2 combinations
4 combinations
5 combinations
6 combinations
A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 2I 2.
The Probability density function for the power P is f(P) = 1/2 * (1/P) for 162 < P < 242.
The power P is equal to 2I^2, so we can find the probability density function of P by finding the distribution of I first. A uniformly distributed random variable X over an interval (a, b) has a probability density function given by:
f(x) = 1/(b - a) for a < x < b
Since I is uniformly distributed over (9, 11), its probability density function is:
f(I) = 1/(11 - 9) = 1/2
Now, to find the distribution of P, we can use the transformation function P = 2I^2:
f(P) = f(I) * |dI/dP|
Using the chain rule, we have:
dI/dP = dI/d(2I^2) * d(2I^2)/dP = 1/2 * (2I) = I/P
So:
f(P) = f(I) * (1/P) = 1/2 * (1/P)
Now, we need to find the bounds for P. The power P can be calculated for any value of I between 9 and 11, so the bounds for P are:
P_min = 2 * 9^2 = 162
P_max = 2 * 11^2 = 242
Therefore, the probability density function for the power P is:
f(P) = 1/2 * (1/P) for 162 < P < 242
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____The given question is incomplete, complete question is given below:
A fluctuating electric current I may be considered a uniformly distributed random variable over the interval (9, 11). If this current flows through a 2-ohm resistor, find the probability density function of the power P = 2I^2.
Solve the formula for h.
S =6pi squared +5pi x r
squared
Therefore, the formula for h, given the assumption that the formula is for the surface area of a cylinder, is: h = 3πr/2 + 3πr²/4.
What is equation?An equation is a mathematical statement that two expressions are equal to each other. It contains an equals sign (=) between two expressions, with one on each side. An equation can be used to describe a relationship between variables, to solve problems, or to represent a mathematical model. Equations can also be more complex and involve multiple variables and operations, such as the quadratic equation: ax² + bx + c = 0, where a, b, and c are constants and x is the variable. Solving equations is an important part of mathematics and many other fields of study, as it allows us to find solutions to problems and to model and understand real-world phenomena.
Here,
The given formula is:
S = 6π² + 5πr²
To solve for h, we need an equation that relates h to S and r. However, there is no h in the given formula. So either the formula is incomplete or we have to assume some relationship between h, S, and r.
If we assume that the formula is for the surface area of a cylinder, then we can relate h to S and r using the formula for the lateral surface area of a cylinder:
L = 2πrh
where L is the lateral surface area, h is the height, and r is the radius.
The total surface area S of a cylinder can be found by adding the area of the two circular bases (2πr²) to the lateral surface area:
S = 2πr² + L
S = 2πr² + 2πrh (substituting L with 2πrh)
Now we can rearrange this formula to solve for h:
S - 2πr² = 2πrh
h = (S - 2πr²) / (2πr)
Substituting the given value of S:
h = (6π² + 5πr² - 2πr²) / (2πr)
h = (6π² + 3πr²) / (2πr)
h = 3πr/2 + 3πr²/4
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Consider a 1 x n checkerboard (1 by n). The squares of the checkerboard are to be painted white and gold, but no two consecutive squares may both be painted white. Let p(n) denote the number of ways to paint the checkerboard subject to this rule (restriction).
Find a recursive formula for p(n) valid for n>=3.
The recursive formula for p(n) is; p(n) = p(n-2) + p(n-3) for n >= 3. Case 1: The last square is painted white. If the last square is painted white, then the second to last square must be painted gold.
There are p(n-2) ways to paint the remaining n-2 squares of the checkerboard subject to the restriction.
Case 2: The last square is painted gold. If the last square is painted gold, then the second to last square can be painted either white or gold. If the second to last square is painted white, then there are p(n-3) ways to paint the remaining n-3 squares of the checkerboard subject to the restriction.
If the second to last square is painted gold, then there are p(n-2) ways to paint the remaining n-2 squares of the checkerboard subject to the restriction.
Therefore, the recursive formula for p(n) is: p(n) = p(n-2) + p(n-3) for n >= 3
with initial conditions p(1) = 2 and p(2) = 3.
The base case for the recursion is p(1) = 2 and p(2) = 3, which are the number of ways to paint a 1 x 1 checkerboard and a 1 x 2 checkerboard subject to the restriction, respectively.
The recursive formula counts the number of ways to paint a 1 x n checkerboard subject to the restriction by considering the last column of the checkerboard.
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