The sequence {r"} converges for |r| < 1 and diverges otherwise
The question requires us to consider the sequence {r"} and determine whether it converges or diverges for different real values of r.
We can use the following test to determine if a series converges or diverges:
If limn→∞ an=0 and an is a decreasing sequence, then the series converges.
If an is not decreasing, then the series diverges.
For the given sequence {r"}, we have:
rn = r × r × r × ... × r (n times) = rn-1 × r
Since rn-1 is a real value, we can see that this sequence is just the geometric sequence with a common ratio of r. For the geometric sequence, the sum of n terms is given by:
S_n = a(1 - rⁿ) / (1 - r)
where a is the first term.
So, if |r| < 1, then the sequence {r"} converges, and if |r| ≥ 1, then the sequence diverges.
Hence, option (A) is the correct choice.
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I need help just did all of this and still lost :D
Answer:
It's the last one...I think... I'm 99.99% sure...
Step-by-step explanation:
Quadrant IV
a color print 21 pages in 7 minutes How many minutes does it take per page?
Answer:
3
Step-by-step explanation:
Cause you would divide 21 by 7 and 7 can only go into 21 3 times
Solve for b
a) 2b x 3 = 6 c) 6 + 7b = 41
b) 32 - 3b = 2 d) 100/ 5b = 2
a) The solution for b in the equation 2b × 3 = 6 is b = 1.
b) The solution for b in the equation 32 - 3b = 2 is b = 10.
c) The solution for b in the equation 6 + 7b = 41 is b = 5.
d) The solution for b in the equation 100/5b = 2 is b = 10.
a) To solve for b in the equation 2b × 3 = 6, we can start by dividing both sides of the equation by 2 to isolate b.
2b × 3 = 6
(2b × 3) / 2 = 6 / 2
3b = 3
b = 3/3
b = 1
Therefore, the solution for b in the equation 2b × 3 = 6 is b = 1.
c) To solve for b in the equation 6 + 7b = 41, we can start by subtracting 6 from both sides of the equation to isolate the term with b.
6 + 7b - 6 = 41 - 6
7b = 35
b = 35/7
b = 5
Therefore, the solution for b in the equation 6 + 7b = 41 is b = 5.
b) To solve for b in the equation 32 - 3b = 2, we can start by subtracting 32 from both sides of the equation to isolate the term with b.
32 - 3b - 32 = 2 - 32
-3b = -30
b = (-30)/(-3)
b = 10
Therefore, the solution for b in the equation 32 - 3b = 2 is b = 10.
d) To solve for b in the equation 100/5b = 2, we can start by multiplying both sides of the equation by 5b to isolate the variable.
(100/5b) × 5b = 2 × 5b
100 = 10b
b = 100/10
b = 10.
Therefore, the solution for b in the equation 100/5b = 2 is b = 10.
In summary, the solutions for b in the given equations are:
a) b = 1
c) b = 5
b) b = 10
d) b = 10
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what is the result of 2.130 x 10³ - 6.6 x 10² =
Answer:
The answer you're looking for is 1470.
Step-by-step explanation:
The method I used was PEMDAS
Since there was no parenthesis, I simplified the exponents.
2.130 x 10³ - 6.6 x 10² = ?
2.130 x 1000 - 6.6 x 100 = ?
After that, I multiplied all terms next to each other.
2.130 x 1000 - 6.6 x 100 = ?
2130 - 660 = ?
The final step I did was to subtract the two final terms and ended up with 1470 as my final answer.
1470 = ?
I hope this was helpful!
By completing the square, work out the coordinate of the turning point of the curve y= x²+ 16x -7
Answer:
(-8,-71)
Step-by-step explanation:
I assume by turning point it means the vertex:
\(y=x^2+16x-7\\y+71=x^2+16x-7+71\\y+71=x^2+16x+64\\y+71=(x+8)^2\\y=(x+8)^2-71\)
Now that we converted our equation to vertex form \(y=(x+h)^2+k\), we can see our vertex, or turning point, is (h,k)=(-8,-71)
Draw a rhombus with diagonals 4 and 6. What is the area of the rhombus?
first of all you have to draw a rhombus
area (d+d)/2
(4+6)/2
5
What is the sum of the measures of the interior angles of an octagon?
Answer:
1080
Step-by-step explanation:
(n-2) × 180
(8-2) × 180= 1080
the u.s.post office will accept a box for shipment only if the sum of height and girth(distance around of bottom) is at most 108 in. find the dimensions of the largest volume of acceptable box with square bottom.
The dimensions of the box of height 36in and the side of the square is 18in.
A topological measure of the size of an object's covering properties is its dimension. It is roughly the number of coordinates required to specify a point on the object.
Given that the u.s.post office will accept a box for shipment only if the sum of height and girth(the distance around of bottom) is at most 108 in.
We must first determine the box's dimensions.
Let s be the side of the square base and h be the height of the box
So , h + 4s = 108
H = 108 – 4s -----(1)
Volume = s^2 h
V = s^2(108-4s)
= 108s2 – 4s3
V’ = 216s – 12s2
Let v’ = 0
S[216-12s] = 0
216 – 12s = 0
12s = 216
S = 216/12
S = 18
V” = 216 – 24s for s =18
V” = 216 – 432
= -216(-ve)
Volume is maximum when s =18
So h= 108-72
=36 in
Therefore the dimensions of the box of height 36in and the side of the square is 18in.
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48 mouse traps/6 people = X mouse traps/1 person
Answer:
8 mousetraps
Step-by-step explanation:
to find how many mousetraps per person, we need to divide 48 by 6, which gives us 8.
EMERGENCY! Please give me the correct answer!
Answer:
6^10
Step-by-step explanation:
6^0 is 1 so 6^10 x 1 is just 6^10
The solid below is similar to a larger solid.
The surface area and volume of the smaller solid is given along
with the scale factor between the two solids. Find the surface area (S) and Volume (V) of the larger solid. Show
all your work.
Surface Area of Larger Solid: S2 = k^2 * S1 and Volume of Larger Solid: V2 = k^3 * V1. Substitute in the given values for S1, V1, and k before solving for S2 and V2.
To solve this problem, we need to use the fact that similar solids have corresponding sides that are proportional in length.
Let's call the scale factor between the two solids "k". This means that if the smaller solid has a side length of "s", then the corresponding side length in the larger solid is "ks".
Now let's use the given information to find the surface area and volume of the larger solid. Let's call the surface area and volume of the smaller solid "S1" and "V1", respectively.
Surface Area:
The surface area of a solid is proportional to the square of its linear dimensions. Since the scale factor between the two solids is "k", the surface area of the larger solid is k^2 times the surface area of the smaller solid. Therefore, the surface area of the larger solid (S2) can be found using the formula:
S2 = k^2 * S1
Volume:
The volume of a solid is proportional to the cube of its linear dimensions. Using the same logic as above, the volume of the larger solid (V2) can be found using the formula:
V2 = k^3 * V1
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-1/5 x 15/16 get 70 pts if u answer thisss tysmmm<3
Answer:
\(\frac{-3}{16}\)
Step-by-step explanation:
\(\frac{-1}{5}\) × \(\frac{15}{16}\)
=> \(\frac{-15}{80}\)
=> \(\frac{-3}{16}\)
Hoped this helped.
find the area (in square feet) of a trapezoid with height $h$h and bases $b_1$b1 and $b_2$b2 . h=6
b1=9
b2=12
The area is
square feet.
Given, the height \($h=6$, the base $b_1=9$ and $b_2=12$.\)
The area of the trapezoid is given by the formula,
\(\[A = \frac{1}{2}h(b_1+b_2)\]\)
Substitute the given values,
\(\[A = \frac{1}{2} \times 6 \times (9+12)\]\\\)
Simplifying the above expression,
\(\[A = 45 \text{ square feet}\]\)
Therefore, the area of the trapezoid is $45$ square feet.
The perimeter of a two-dimensional geometric shape is the entire length of the boundary or outer edge. It is the sum of the lengths of all the shape's sides or edges.
The perimeter of a square, for example, is calculated by adding the lengths of all four sides of the square.
Similarly, to find the perimeter of a rectangle, sum the lengths of two adjacent sides and then double the result because there are two pairs of adjacent sides.
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solve for y. 9=5-y/3
Answer:
y = -1.3 or y = -22
Step-by-step explanation:
5-y/3=9
5/3 9/3
1.66666666667 = 1.7
1.7-y=3
-y/-1=1.3/-1
y=-1.3
or
5-y/3=9
5-y=27
5=27+y
5-27=y
-22=y
Find the value of x
Answer:
A) 33
Step-by-step explanation:
180 - 105= 75
75 + 72 + x = 180
147 + x = 180
x= 33
Which expression is not equivalent to (5^2x)^3
Answer:
#3
Step-by-step explanation:
For this... you just multiply the exponents together to get 6x
any of the others that do NOT have 6x as a product of exponents is not equiv ..... so #3 is not equivalent
Option 3 is not equivalent to the given expression.
What is an equivalent expression?Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.
The given expression is (5²ˣ)³.
Now, (5⁶ˣ)
(5ˣ)⁶
(5²)³ˣ
Therefore, option 3 is not equivalent to the given expression.
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In parallelogram ABCD, AB = 7x + 6, BC = 2x + 16, CD = 3x + 10. What is the length of AD?
O 1
O 13
O 18
O 20
graph the equation using slope-intercept form.
y=3/4x-1
where would i place the dot in the graph ?
Can someone heeeeelp???
PLZ HELP IM TIMMED 50 POINTS The total sales for June at Jim’s Candy Store were $7,785. The total sales for June and July
were $12,603. What were the total sales for July?
Answer:
the answer is 4818 pls give me brainliest
Step-by-step explanation:
if the dot product of two nonzero vectors is zero, the vectors must be perpendicular to each other. a) true b) false
The statement "if the dot product of two nonzero vectors is zero, the vectors must be perpendicular to each other" is true. The dot product of two vectors is zero if and only if the vectors are perpendicular.
The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them. When the dot product is zero, it means that the cosine of the angle between the vectors is zero, which occurs when the vectors are perpendicular.
In other words, the dot product being zero indicates that the vectors are at a 90-degree angle to each other, supporting the statement that they are perpendicular.
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snowdon has a height of approximately 1100 metres above the sea
assuming that the temperature decreases by 1 oc for every 100m above sea level, work out the temperature at the summit of snowdon when the sea level temperature is 4oc
Answer:
\(-7^\circ C\)
Step-by-step explanation:
Let
x -----> the height in meters
y ----> the temperature in Celsius degree
we know that
The linear equation in slope intercept form is equal to
\(y+mx+b\)
where
m is the slope or unit rate
b is the y-intercept or initial value
In this problem we have
The slope or unit rate is equal to
\(m=-\dfrac{1}{100} \dfrac{^\circ C}{m}\) ---> is negative because is a decreasing function
The y-intercept or initial value is equal to
\(b=4^\circ C\) ----> value of y when the value of x is equal to zero (sea level)
substitute
\(y=-\dfrac{1}{100} x+4\)
For x=1,100 m
substitute in the linear equation and solve for y
\(y=-\dfrac{1}{100} (1,100)+4\)
\(x=-7^\circ C\)
Solve the equation for the specified variable. Y=AB-2, For A
Answer:
b = 1
b = -1
a = 0
Step-by-step explanation:
What is A in degrees?
Sin (A)= 0.7721
Answer:
angle
Step-by-step explanation:
it means angle. no prob
2, 10, 50, 250, 1250
Write the explicit form of the sequence above.
The explicit function of the sequence is a(n) = 2(5)^n-1
How to determine the explicit function of the sequenceThe definition of the function is given as
2, 10, 50, 250, 1250
The above definitions imply that we simply multiply 5 to the previous term to get the current term
Using the above as a guide,
So, we have the following representation
a(1) = 2
Common ratio, r = 5
So, we have
a(n) = a * r^n-1
This givs
a(n) = 2(5)^n-1
Hence, the sequence is a(n) = 2(5)^n-1
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Alicia and Leah are making cookies for a fundraiser.
Leah made 10% more cookies than Alicia.
They are selling the cookies for $1.00 each.
They sold 90% of the cookies they made.
Write an equation to represent the total amount of money
Find all EXACT solutions of the equation given below in the interval \( [0, \pi) \). \[ \sin (3 x)=-\frac{\sqrt{3}}{2} \] If there is more than one answer, enter them in a list separated by commas. En
The required exact solutions of this equation are \($$\boxed{\frac{4\pi}{9}, \frac{5\pi}{9}, \frac{16\pi}{9}, \frac{17\pi}{9}}$$\)
The given equation is
\($\sin(3x)=-\frac{\sqrt{3}}{2}$.\)
The first step to solving this equation is to solve for \($3x$\).
We know that
\($\sin(60^o) = \frac{\sqrt{3}}{2}$,\) so we need to find the angle whose sine is
\($-\frac{\sqrt{3}}{2}$\) (since $\sin$ is negative in the third and fourth quadrants).
This angle will be \($240°$\) since \($\sin(240^o) = -\frac{\sqrt{3}}{2}$\).
The reference angle for $240°$ is $60°$, which is the same as the reference angle for \($\frac{\sqrt{3}}{2}$\).
Since the sine function is negative in the third and fourth quadrants, we must add $180°$ to each solution to get the angles in the interval $[0, \pi)$.
Hence, we have:
\($$\begin{aligned} 3x&=\frac{4\pi}{3}+360^on\\ 3x&=\frac{5\pi}{3}+360^om \end{aligned}$$\)
where $n, m$ are any integer.
Find exact solutions by solving for \($x$\) in each equation.
We get: \($$\begin{aligned} x&=\frac{4\pi}{9}+120^on\\ x&=\frac{5\pi}{9}+120^om \end{aligned}$$\)
where $n, m$ are any integer.
Since the interval is\($[0, \pi)$\), we only need to consider the values of \($[0, \pi)$\) and \($m$\) that make \($x$\) in this interval.
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The exact solution is \($x=\frac{2\pi}{9}$\) (in radians). The required solution is: \($\frac{2\pi}{9}$\).
The given equation is:
\($ \sin (3 x)=-\frac{\sqrt{3}}{2} $\)
The interval is \($[0, \pi)$\)
To solve for x, use inverse sine function on both sides:
\(\[\begin{aligned}\sin (3 x)&=-\frac{\sqrt{3}}{2} \\ \sin^{-1} \sin (3 x)&=\sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \\ 3 x &= -\frac{\pi}{3} + k \pi \quad \text{or} \quad 3 x = \frac{2\pi}{3} + k \pi, \quad \text{where} \quad k\in \mathbb{Z}\end{aligned}\]\)
To get the values of x in the interval \($[0, \pi)$\):
For
\($3x = -\frac{\pi}{3}$\)
we have \($x = -\frac{\pi}{9}$\),
which is outside the given interval.
For \($3 x = \frac{2\pi}{3}$\),
we have \($x = \frac{2\pi}{9}$\),
which is within the given interval.
So, the exact solution is \($x=\frac{2\pi}{9}$\) (in radians).
Therefore, the required solution is: \($\frac{2\pi}{9}$\).
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Given that g(4)=9 and g(-11)=k, find the value of k that would result in a slope of 5 between the two points.
Answer:
\(k=-66\)
Step-by-step explanation:
So we are given two values:
\(g(4)=9 \text{ and } g(-11)=k\)
They can be interpreted as: (4,9) and (-11,k).
So, we want to find the value of k such that the slope of the line between the two points would be 5.
Recall the slope formula. It is:
\(m=\frac{y_2-y_1}{x_2-x_1}\)
Let (4,9) be x₁ and y₁ and let (-11,k) be x₂ and y₂. Substitute 5 for m. Therefore:
\(5=\frac{k-9}{-11-4}\)
Simplify the denominator:
\(5=\frac{k-9}{-15}\)
Multiply both sides by -15. The right side cancels:
\(-15(5)=(-15)\frac{k-9}{-15}\\ -75=k-9\)
Now, add 9 to both sides. The right side cancels.
\((-75)+9=(k-9)+9\\k=-66\)
Therefore, k is -66.
What are the solutions for the equation 2cos0+1=0 ?
Answer:
2cos0=-1
cos0=-1/2
proved
determine the critical values for a two-tailed test (h1: μ ≠ μ0) of a population mean at the α = 0.05 level of significance based on a sample size of n = 12.
The critical values for this test are -2.201 and 2.201. If your test statistic falls outside this range, you would reject the nullSo, the critical values for this test are -2.201 and 2.201. If your test statistic falls outside this range, you would reject the null hypothesis in favor of the alternative hypothesis (H1: μ ≠ μ0). in favor of the alternative hypothesis (H1: μ ≠ μ0).
To determine the critical values for a two-tailed test with a sample size of n = 12 and a significance level of α = 0.05, we need to consult a t-distribution table.
First, we need to find the degrees of freedom, which is equal to n - 1 = 12 - 1 = 11.
Next, we look up the t-value for a two-tailed test with a 0.025 level of significance (0.05/2) and 11 degrees of freedom. From the t-distribution table, we find that the t-value is 2.201.
Therefore, the critical values for a two-tailed test of a population mean at the α = 0.05 level of significance based on a sample size of n = 12 are -2.201 and +2.201.
This means that if our calculated t-value falls outside of this range, we can reject the null hypothesis (H0: μ = μ0) in favor of the alternative hypothesis (H1: μ ≠ μ0) at the 0.05 level of significance.
To determine the critical values for a two-tailed test of a population mean at the α = 0.05 level of significance with a sample size of n = 12, you'll need to use a t-distribution table.
Since it's a two-tailed test, you'll need to find the t-score that corresponds to α/2, which is 0.025 in each tail. With a sample size of 12, you have 11 degrees of freedom (df = n - 1).
Looking up the t-distribution table with df = 11 and α/2 = 0.025, you'll find the critical t-value to be approximately ±2.201.
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