The pοpulatiοn οf the tοwn in the year 1995 was apprοximately 4,262.
Hοw tο determine the percentage grοwth rate?The grοwth factοr (b) is given as: b = 3.76
The grοwth factοr (b) is greater than 1.
We can sοlve this prοblem by using the fοrmula fοr expοnential grοwth:
[tex]P(t) = P0(1 + r)^t[/tex]
where:
P0 is the initial pοpulatiοn (821 in 1970)
r is the annual grοwth rate (11% οr 0.11)
t is the number οf years since the initial pοpulatiοn (25 years frοm 1970 tο 1995)
Substituting the given values, we get:
[tex]P(25) = 821(1 + 0.11)^25\\\\P(25) = 821(1.11)^25[/tex]
Using a calculatοr tο evaluate this expressiοn:
P(25) ≈ 4,261.9
Therefοre, the pοpulatiοn οf the tοwn in the year 1995 was apprοximately 4,262.
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Mateo y leo caminan para entrenar y encontrarse en un punto. Mateo camina 1/5 mas de lo que camina leo. Si entre los dos caminan 15Km, ¿ cuantos kilómetros recorre cada uno?
Answer:
Primero, podemos establecer las siguientes ecuaciones:
La suma de los kilómetros que caminan Mateo y Leo es igual a 15:
Mateo + Leo = 15
Mateo camina 1/5 más que Leo:
Mateo = Leo + 1/5Leo
Mateo = 6/5Leo
Podemos utilizar la segunda ecuación para despejar una de las variables en términos de la otra. Por ejemplo, despejando Leo:
Leo = 5/6Mateo
Luego, podemos sustituir esta expresión en la primera ecuación para obtener una ecuación con una sola variable:
Mateo + 5/6Mateo = 15
11/6Mateo = 15
Mateo = 15*6/11
Mateo = 8.18 Km
Finalmente, podemos utilizar una de las ecuaciones que relaciona a Mateo y Leo para obtener la distancia que recorre Leo:
Leo = 5/6Mateo
Leo = 5/6*8.18
Leo = 6.82 Km
Por lo tanto, Mateo recorre 8.18 Km y Leo recorre 6.82 Km.
Which of the following is the same as 8(9+2)?
8*9+8
8*9+2
8*9+8*2
9(8+2)
2(8+9)
Answer:
The third one
Step-by-step explanation:
if you look at it, and actually solve it its 88.
(1x10⁶) - (8x10⁴) I'm scientific notation
Answer: 9.2 x 105
Step-by-step explanation:
l-80-20l – 100 + l-100l = ???
Step-by-step explanation:
We can simplify the given expression using the rules of absolute values:
l-80-20l – 100 + l-100l
= -(20l + 80) - 100 + 100 (since |x| = -x when x is negative)
= -20l - 80
= -20(l + 4)
Therefore, the simplified expression is -20(l + 4).
Answer:
100
Step-by-step explanation:
[tex] | - 80 - 20| - 100 + | - 100| \\ = | - 100| - 100 + | - 100| \\ = 100 - 100 + | - 100| \\ = | - 100| \\ = \boxed{100}[/tex]
____________
hope it helps!
intelligence test scores refered to as intelligent quotient or IQ scores are based on characteristics such as verbal skills,abstruct reasoning power, numerical ability and spatial visualization.if plotted on a graph the distribution of IQ scores approximates a normal curve with a mean of about 100. an IQ scores above 115 is considered superior studies of "intellectually gifted" children have generally defined the lower limit of their IQ scores at 140: approximately 1% of the population have IQ scores above this limit.find the standard deviation of this distribution?
The standard deviation of the distribution is given as follows:
[tex]\sigma = 17.2[/tex]
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The mean of the distribution of IQ scores is given as follows:
[tex]\mu = 100[/tex]
X = 140 is the 99th percentile, as approximately 1% of the population have IQ scores above this limit, hence when X = 140, Z = 2.327, meaning that the standard deviation is obtained as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.327 = \frac{140 - 100}{\sigma}[/tex]
[tex]2.327\sigma = 40[/tex]
[tex]\sigma = \frac{40}{2.327}[/tex]
[tex]\sigma = 17.2[/tex]
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Work 1.1.8 The president of South Africa, Mr Cyril Ramaphosa, announced that there will be an inflow of R10 billion investment in South Africa in the following year. Use the multiplier formula to calculate the eventual change in aggregate expenditure due to this R10 billion injection. The marginal propensity to save (mps) is 0,4. Remember to show all formulas and calculations!
Using the multiplier formula, the eventual change in aggregate expenditure due to the inflow of R10 billion into South Africa and using the marginal propensity to consume (MPC) is R6 billion.
What is the multiplier formula?The multiplier formula is used to compute the amount of new income generated from the flow of extra income.
The multiplier formula uses the MPS (marginal propensity to save) to compute the multiplier as follows:
M = 1 / MPS
The marginal propensity to save gives the proportion of income that will be saved.
The marginal propensity to save (MPS) = 0.4
M = 1 / MPS
M = 1/0.4)
= 2.5
Change in aggregate expenditure = R25 billion (2.5 x R10 billion)
Thus, we expect the aggregate expenditure in South Africa to increase by R25 billion following the inflow of R10 billion investment.
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1. What is a reasonable distance between two cities? (1 point) 200 km 200 m 200 cm 200 mm
Answer:
200km
Step-by-step explanation:
Well.. if 2 cities were 200m apart, 200 cm apart, or 200 mm apart, they would be considered one city. The only logical choice is 200km.
Answer:
a
Step-by-step explanation:
1. Write the equation of the circle, (x-2)²+(y-5)²=7² in general form.
Answer:
The equation in general form is x² + y² - 4x - 10y - 20 = 0
Answer:
x² + y² - 4x - 10y - 20 = 0
Step-by-step explanation:
The general equation of circle is
x² + y² + ax + by + c = 0
where a, b and c are constants
To convert the standard form of the equation (x - 2)² + (y - 5)² = 7² into general form, expand the squares and the constant on the right side and adjust the terms to have 0 on the right side
(x - 2)² = x² - 4x + 4
(y - 5)² = y² - 10y + 25
7² = 49
(x - 2)² + (y - 5)² = 7²
= x² - 4x + 4 + y² - 10y + 25 = 49
Subtract 49 from each side to get 0 on the right:
x² - 4x + 4 + y² - 10y + 25 - 49 = 0
Simplify the constant terms
4 + 25 - 49 = 29 - 49 = -20
Required equation is
x² + y² - 4x - 10y - 20 = 0
The following table represents measurement of the height of a bean sprout:
Days after sprout
was planted 7 9 11 13
Height of sprout
in inches 3 4.5 6 7.5
Create a linear equation to represent this situation. According to your equation, how tall will the sprout
be after 14 days?
The linear equation representing the situation is y = 0.75x - 2.25, and according to this equation, the height of the sprout after 14 days would be 8.25 inches.
To create a linear equation that represents the relationship between the height of the bean sprout and the number of days after planting.
Therefore, the slope of the line is:
slope = change in height / change in days = (7.5 - 3) / (13 - 7) = 1.5
The intercept of the line can be found by substituting one of the data points into the point-slope form of a line:
y - y1 = m(x - x1)
where m is the slope, x1 and y1 are the coordinates of one of the data points. We choose the first data point (7, 3):
y - 3 = 1.5(x - 7)
y - 3 = 1.5x - 10.5
y = 1.5x - 7.5
Therefore, the linear equation that represents the relationship between the height of the bean sprout (y) and the number of days after planting (x) is:
y = 1.5x - 7.5
To find the height of the sprout after 14 days, we substitute x = 14 into the equation:
y = 1.5(14) - 7.5
y = 10.5
Therefore, the height of the sprout after 14 days is predicted to be 10.5 inches.
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 In a triangle, the second angle measures two times the first. The measure of the third angle is 170° more than the measure of the second. Find the measures of the three angles
Answer: Let's call the first angle "x".
From the problem statement, we know that the second angle is two times the first, so:
Second angle = 2x
We also know that the third angle is 170° more than the second angle, so:
Third angle = Second angle + 170°
Third angle = 2x + 170°
We know that the sum of the three angles in a triangle is 180°, so we can set up an equation:
x + 2x + (2x + 170°) = 180°
Simplifying and solving for x:
5x + 170° = 180°
5x = 10°
x = 2°
So the first angle is 2°. Using this value, we can find the other two angles:
Second angle = 2x = 2(2°) = 4°
Third angle = 2x + 170° = 2(2°) + 170° = 174°
Therefore, the three angles in the triangle measure 2°, 4°, and 174°.
Step-by-step explanation:
Solve by factoring (15 points)
g²+4g-32=0
A 9=32 or 28
B 9=4 or 8
C 9=2 or -16
D 9=4 or -8
[tex]g^{2} +4g-32=0[/tex]
[tex]g^{2} + 8g - 4g - 32=0[/tex]
[tex]g(g+8)-4(g+8) = 0[/tex]
[tex](g+8)(g-4) = 0[/tex]
[tex]g+8=0 \implies \bf g = -8[/tex]
[tex]g-4=0 \implies \bf g = 4[/tex]
[tex]\implies g = 4 \ or \ -8[/tex]
Vanessa borrowed $5,000 at 4% simple interest and will pay after 2 years. Please help!!
a) How much will she have to pay?
b) How much interest will she pay?
Answer:
a) To calculate how much Vanessa will have to pay back after 2 years, we need to add the interest to the principal.
The interest can be calculated using the simple interest formula:
Interest = Principal x Rate x Time
where,
Principal = $5,000 (the amount borrowed)
Rate = 4% (expressed as a decimal, 0.04)
Time = 2 years
So,
Interest = $5,000 x 0.04 x 2
Interest = $400
The total amount Vanessa will have to pay back is the sum of the principal and the interest:
Total amount = Principal + Interest
Total amount = $5,000 + $400
Total amount = $5,400
Therefore, Vanessa will have to pay $5,400 after 2 years.
b) To calculate how much interest Vanessa will pay, we can use the same formula for simple interest:
Interest = Principal x Rate x Time
Plugging in the given values, we get:
Interest = $5,000 x 0.04 x 2
Interest = $400
Therefore, Vanessa will pay $400 in interest over the 2-year period.
please answer the question
a.) The area covered by algae for 10 days after it was spotted would be = 35.7 m²
How to calculate the area covered by the algae after 10 days?The area covered by the algae when it was first spotted = 12.5 m²
For every week(7 days) = 2(12.5)
But 10 days = x
Make X the subject of formula;
X = 10× 2(12.5)/7
X = 250/7
X = 35.7 m²
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How do I correctly simplify the expression and where did the student go wrong?
Part A: The student that simplified the give trigonometrical identity expression incorrectly is student B.
Part B: The correct complete solution is shown below
Simplifying Trigonometrical identities expressionsFrom the question, we are to determine the student that simplified the expression incorrectly.
Part A:
The student that simplified the expression incorrectly is student B.
The student is went wrong in step 3.
From step 2, the student divided (1 + tan²θ) by (tanθ/cotθ). The student is supposed to divide [(1 + tan²θ)/tanθ] by cotθ
Part B:
The correct solution is shown below
[(1 + tan²θ)/tanθ] ÷ cotθ
(sec²θ/tanθ) ÷ cotθ
Which can be written as:
(sec²θ/tanθ) ÷ 1/tanθ
Then,
(sec²θ/tanθ) × tanθ/1
= sec²θ/1
= sec²θ
Hence, the correct simplified expression is sec²θ
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NEED HELP!!
Select which points are solutions to this inequality.
y ≥ 5x - 7
(-5, -3)
(-4, 2)
(1, -2)
(-1, 0)
(3, 0)
(4, 1)
Inequality y 5x - 7 has solutions (-5, -3), (-4, 2), and (-1, 0).
What are the three potential remedies for the inequality?In order to address an inequality • You can multiply or divide each side by the same positive amount; • You can add the same amount to each side; • You can take the same amount away from each side. If you multiply or divide each side by a negative value, the inequality sign must be reversed.
Let's examine each point individually:
(-5, -3)
y ≥ 5x - 7
-3 ≥ 5(-5) - 7
-3 ≥ -32
The inequality is true, so (-5, -3) is a solution.
(-4, 2)
y ≥ 5x - 7
2 ≥ 5(-4) - 7
2 ≥ -27
The inequality is true, so (-4, 2) is a solution.
(1, -2)
y ≥ 5x - 7
-2 ≥ 5(1) - 7
-2 ≥ -2
The inequality is not true, so (1, -2) is not a solution.
(-1, 0)
y ≥ 5x - 7
0 ≥ 5(-1) - 7
0 ≥ -12
The inequality is true, so (-1, 0) is a solution.
(3, 0)
y ≥ 5x - 7
0 ≥ 5(3) - 7
0 ≥ 8
The inequality is not true, so (3, 0) is not a solution.
(4, 1)
y ≥ 5x - 7
1 ≥ 5(4) - 7
1 ≥ 13
The inequality is not true, so (4, 1) is not a solution.
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Draw a circle with radius 4cm.
What is the length of the diameter of the circle?
How many sectors of 90° will fit inside the circle?
Draw five radii inside your circle that are equally spaced out around the circumference. Join up the ends of the radii to create a shape inside your circle.
What is the name of the shape that you have created inside your circle?
How long are the chords that are joining the radii together?
How big is the angle of each sector?
If you did the same as above with sectors of 45°, what shape would you create inside your circle?
The length of the diameter of the circle of the radius 4 cm is 8 cm and other solutions are added below
A circle with radius 4cm.See attachment for the circle
The length of the diameter of the circleThis is calculated as
Diameter (d) = 2 * Radius (r)
So, we have
d = 2 * 4 cm
d = 8 cm
Number of sectors of 90°The number of 90° sectors that will fit inside the circle is
Sectors = (360°/90°)
Sectors = 4
Five radii inside the circle equally spacedSee attachment for the circle
The name of the shape in the circleThe shape created inside the circle by joining the ends of five equally spaced radii is a regular pentagon.
Length of the chords joining the radii togetherThe length (L) of one chord is
L = 2 × r × sin(c/2)
For 6 chords created by the 5 radii, we have
Total length = 6 × 2 × r × sin(c/2)
Where
c = 360/6 = 60
So, we have
Total length = 6 × 2 × 4 × sin(60/2)
Evaluate
Total length = 24 cm
How big is the angle of each sector?Each sector of 90° in a circle has an angle of
Angle = (360/6)°
Angle = 60°
What shape would you create inside your circle?If sectors of 45° were used, the shape created inside the circle by joining the ends of the radii would be a nonagon
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Help, please? lol its for like homework or whatev. helpppp!!
Ivan bought a desk on sale for $179.20. This price was 32% of the original price.
Answer:
Below
Step-by-step explanation:
I suppose you are looking for the ORIGINAL price
x = original
32% is .32 in decimal
so .32 * x = $ 179.20
x = 179.20/.32 = $ 560.
I REALLY REALLY REALLY NEED HELP PLS
The answer of the given question based on Statistics to find minimum monthly premium payments is option D) Plan 2, since it has lower monthly premium payments.
What is Probability?Probability is measure of likelihood or chance of an event occurring. It is expressed as number between 0 and 1, where 0 indicates an impossible event and 1 indicates certain event
Assuming that the coverage details are similar, we can compare the monthly premium payments for each plan based on the different categories of coverage:
Individual: Plan 1 has a monthly premium of $87.32, while Plan 2 has a monthly premium of $11.77. Therefore, Plan 2 has the lower monthly premium for this category of coverage.
Individual+Spouse: Plan 2 has a monthly premium of $735.00, while Plan 1 has a monthly premium of $890.00. Therefore, Plan 2 has the lower monthly premium for this category of coverage.
Individual+Child: Plan 2 has a monthly premium of $606.00, while Plan 1 has a monthly premium of $750.00. Therefore, Plan 2 has the lower monthly premium for this category of coverage.
Individual+Family: Plan 2 has a monthly premium of $1,400.00, while Plan 1 has a monthly premium of $1,600.00. Therefore, Plan 2 has the lower monthly premium for this category of coverage.
Based on these comparisons, we can see that Plan 2 has a lower monthly premium for all categories of coverage. Therefore, the answer is option D: Plan 2, since it has lower monthly premium payments.
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Jason wants to buy a new HD television but he thinks that if he waits, the quality of HD televisions will improve. The television he wants to buy costs $2500 now, and based on pricing trends, Jason thinks that the price will increase by 4% each year. (SHOW ALL WORK)
a. Write an exponential growth equation to represent the price y of a new HD television t years from now.
b. Use the equation to predict when a new HD television will cost $3000.
c. Jason decides to wait to buy a new television and saves his money. He puts $2200 in a savings account with 4.7% annual interest compounded continuously. Determine when the amount in his savings will exceed the cost of a new television.
the amount in Jason's savings will exceed the cost of a new television approximately 7.22 years from now.
a. The exponential growth equation to represent the price y of a new HD television t years from now is:
y = $2500[tex](1 + 0.04)^t[/tex]
where t is the number of years from now.
b. To predict when a new HD television will cost $3000, we can set y = $3000 and solve for t:
$3000 = $2500[tex](1 + 0.04)^t[/tex]
1.2 = [tex]1.04^t[/tex]
ln(1.2) = t ln(1.04)
t = ln(1.2) / ln(1.04)
t ≈ 3.97
Therefore, a new HD television will cost $3000 approximately 4 years from now.
c. To determine when the amount in Jason's savings will exceed the cost of a new television, we can use the formula for continuous compound interest:
A = [tex]Pe^(rt)[/tex]
where A is the amount in savings after t years, P is the principal amount, e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate expressed as a decimal, and t is the time in years.
In this case, P = $2200, r = 0.047 (since the annual interest rate is 4.7%), and we want to find the time t when A = $2500. Plugging these values into the formula, we get:
$2500 = $2200[tex]e^(0.047t)1.13636 = e^(0.047t)[/tex]
ln(1.13636) = 0.047t
t = ln(1.13636) / 0.047
t ≈ 7.22
Therefore, the amount in Jason's savings will exceed the cost of a new television approximately 7.22 years from now.
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Please help will mark Brainly
For the function g, the vertex form equation is [tex]g(x) = -2(x - 5)^{2} + 9.[/tex]
What sort of equation would that be?In algebra, the idea of an equation is a logical assertion that shows the equivalence of several mathematical expressions. For instance, the equations 3x + 5 & 14, that are separated by the 'same' sign, make up the equation 3x + 5 = 14.
What about the meaning of the equation?A mathematical equation, such as 6 × 4 = 12 x 2, states that two quantities or values are equivalent. a meaningful noun. Equations are used when more than one factor has to be considered in order to fully understand or explain a situation.
Substituting the first point (7,1) into the equation for g(x), we get:
[tex]1 = a(7 - 5)^{2} + 9[/tex]
we get
[tex]-8 = 4a[/tex]
Dividing both sides by 4,
[tex]a = -2[/tex]
Now that we have found the value of a, we can write the equation for g(x) as:
[tex]g(x) = -2(x - 5)^{2} + 9[/tex]
Therefore, the equation in vertex form for the function g is [tex]g(x) = -2(x - 5)^{2} + 9[/tex].
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There are 3 teaspoons in a tablespoon. There are 10 16 times more teaspoons in a cup. How many teaspoons are in a cup?
which of the following statements are not true? check all that apply.
Answer:
A
b
D
Step-by-step explanation:
The sun revolves around the Earth." Statements 1 and 4 are true.
As the question asks for statements that are not true, I will go through each statement and determine if it is true or false. Statement 1: "Water boils at 100 degrees Celsius."
This statement is true. Water boils at 100 degrees Celsius at sea level under normal atmospheric conditions.
Statement 2: "The moon is made of cheese." This statement is not true. The moon is not made of cheese. It is primarily composed of rock and does not contain any dairy products.
Statement 3: "The sun revolves around the Earth." This statement is not true.
The Earth revolves around the sun in an elliptical orbit. The concept that the Earth is at the center of the solar system and the sun revolves around it was disproven by the heliocentric model proposed by Nicolaus Copernicus in the 16th century.
Statement 4: "Gravity only exists on Earth." This statement is not true. Gravity exists everywhere in the universe. It is a fundamental force that attracts objects with mass towards each other.
While the strength of gravity may vary depending on the mass of the objects and their distance apart, it is present throughout the universe.
To summarize, the statements that are not true are: 2. "The moon is made of cheese." 3. "The sun revolves around the Earth." Statements 1 and 4 are true.
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Need help on multiple choice question
Write two numbers that multiply to the value on top and add to the value on bottom.
Submit Answer
42
-13
Answer:
-6 and -7
Step-by-step explanation:
-6 + -7 = -13
-6 * -7 = 42
ASAP HELP!! I REALLY NEED HELP
Answer:
X = 18
Step-by-step explanation:
7x is the opposite angle if 3x, so 10x= 180 180÷10 = 18
a car travels x km in 3 hours how far will it travel in 9 hours?
Answer:
distance traveled in 1 hour = x/3
To find the distance traveled in 9 hours, we can multiply the distance traveled in 1 hour by 9:
distance traveled in 9 hours = (x/3) * 9
Simplifying this expression gives:
distance traveled in 9 hours = 3x
Therefore, the car will travel 3 times the distance it travels in 3 hours when it travels for 9 hours.
One large jar and two small jars together can hold 8 ounces of jam. One large jar minus one small jar can hold 2 ounces of jam.
A matrix with 2 rows and 2 columns, where row 1 is 1 and 2 and row 2 is 1 and negative 1, is multiplied by matrix with 2 rows and 1 column, where row 1 is l and row 2 is s, equals a matrix with 2 rows and 1 column, where row 1 is 8 and row 2 is 2.
Use matrices to solve the equation and determine how many ounces of jam are in each type of jar. Show or explain all necessary steps.
Please help me solve it through matrices, not by equations. There's a certain way to solve this, I'm just confused on this specific way by using matrices.
There are 2 ounces of jam in the large jar and 2/3 ounce of jam in each small jar.
Describe Matrix?In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are commonly used to represent and manipulate linear transformations and systems of linear equations.
The elements of a matrix are typically denoted by lowercase letters with subscripts, such as aij or bij, where i and j refer to the row and column indices, respectively. For example, a 3x3 matrix might look like this:
| a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
This matrix has three rows and three columns, and the element in the first row and second column is denoted by a12.
Matrices can be added, subtracted, multiplied, and divided, and these operations follow specific rules that are defined in matrix algebra. For example, to add or subtract two matrices of the same size, we simply add or subtract the corresponding elements. To multiply two matrices, we multiply the rows of the first matrix by the columns of the second matrix and sum the products.
Let's represent the number of ounces of jam in the large jar as "L" and the number of ounces of jam in each small jar as "S". Then we can write the system of equations:
1L + 2S = 8
1L - 1S = 2
We can write this system in matrix form as:
[1 2; 1 -1][L;S] = [8;2]
To solve for L and S, we can use matrix algebra to isolate [L;S]:
[L;S] = [1 2; 1 -1]⁻¹ [8;2]
First, we need to find the inverse of the coefficient matrix [1 2; 1 -1]. The inverse is:
[1 -2; 1 1]/3
So we can substitute this into our equation:
[L;S] = [1 -2; 1 1]/3 [8;2]
[L;S] = [2;2/3]
So there are 2 ounces of jam in the large jar and 2/3 ounce of jam in each small jar.
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The complete question is:
SOMEONE PLS HELP ILL MARK BRAINLIEST
Answer:
see explanation below
Step-by-step explanation:
4. ∠A = 35, a = 8.9, find c
since side a is opposite to m∠A and c is the hypotenuse, we can use sin(∠A) = opp/hypo equation
5. sin(35) = 8.9/c
=> c = 8.9/sin(35) = 8.9/0.5736 = 15.516
6. 15.5
I will give brainliest to whoever answers this correctly, Answer is not x=0.1
The value of x in the infinite geometric series, x = (7 ± i√107) / 26.
What is the value of xWe can start by simplifying the left-hand side of the equation:
-1 + 1/x + x + x² + ...+ xⁿ + ...
= -1 + (1/x) * (1/(1 - x))
Using the formula for the sum of an infinite geometric series:
1 + r + r² + ... = 1/(1 - r) for |r| < 1
We can simplify the left-hand side further:
= -1 + (1/x) * (1/(1 - x))
= -1 + (1/(x - x²))
Now we have:
-1 + (1/(x - x²)) = 10/3
Multiplying both sides by 3(x - x²), we get:
-3(x - x²) + 3 = 10(x - x²)
Simplifying, we get:
13x² - 7x + 3 = 0
Using the quadratic formula, we can solve for x:
x = (7 ± √(7² - 4133)) / (2*13)
x = (7 ± i√(107)) / 26
Since the expression is defined only for |x| < 1, we can discard the solution with the plus sign, leaving:
x = (7 - √(-143)) / 26
x = (7 - 3i√13) / 26
Therefore, the solution for x is x = (7 ± i√107) / 26.
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