By the principle of mathematical induction, we have shown that for all positive integers n, n^3 + 2n is divisible by 3.
Proof by Induction: n^3 + 2n is divisible by 3 for all positive integers n.
Base Case:
For n = 1, n^3 + 2n = 1^3 + 2(1) = 3, which is divisible by 3.
Induction Hypothesis:
Assume that for some positive integer k, k^3 + 2k is divisible by 3. We can write this as: k^3 + 2k = 3m, where m is an integer.
Induction Step:
We need to show that (k + 1)^3 + 2(k + 1) is divisible by 3. Expanding the expression:
(k + 1)^3 + 2(k + 1) = k^3 + 3k^2 + 3k + 1 + 2k + 2
= (k^3 + 2k) + (3k^2 + 3k + 3)
Substituting the induction hypothesis:
= 3m + 3(k^2 + k + 1)
= 3(m + k^2 + k + 1)
Since k^2 + k + 1 is an integer (sum of three integers), and m is an integer, their sum (m + k^2 + k + 1) is also an integer. Therefore, (k + 1)^3 + 2(k + 1) is divisible by 3.
What is the r value of the following geometric sequence? 4, 7, 12.25, 21.4375.
Answer:
r ( common ratio) = 1.75
Step-by-step explanation:
Given : geometric sequence 4, 7, 12.25, 21.4375.
To find : What is the r value .
Solution : We have given
Geometric sequence 4, 7, 12.25, 21.4375.
r ( common ratio) = [tex]\frac{second\ term}{first\ term}[/tex].
r ( common ratio) = [tex]\frac{7}{4}[/tex] = 1.75
r ( common ratio) = [tex]\frac{12.25}{7}[/tex] = 1.75
Therefore, r ( common ratio) = 1.75
CD is perpendicular to AB and passes through point C (5,12).
If the coordinates of A and B are (-10, -3) and (7, 14), respectively, the x- intercept of CD is:
(12, 0)
(15, 0)
(17, 0)
or (19,0).
The point:
(-5, 24)
(-2, 19)
(7, -10)
or (8, 11)
lies on CD.
Answer
CD is 17,0
The point -2,19
Step-by-step explanation:
solve this problem 2-(-8)+(-3)=
Answer:
7Step-by-step explanation:
(-)(-) = (+)
(-)(+) = (+)(-) = (-)
(+)(+) = (+)
2 - (-8) + (-3) = (*)
-(-8) = 8
+(-3) = -3
(*) = 2 + 8 - 3 = 10 - 3 = 7
In rectangle ABCD, BC=30 and AB=40. If diagonals AC and BD intersect at E, find the measure of BE.
To find the measure of BE in rectangle ABCD, we can use the properties of a rectangle and similar triangles. By using the Pythagorean theorem, we find the length of the diagonal AC. Then, we use the properties of similar triangles to calculate the length of BE.
Explanation:To find the measure of BE, we can use the properties of a rectangle. In rectangle ABCD, the diagonals are equal in length and bisect each other. Therefore, we can use the Pythagorean theorem to find the length of the diagonal AC: AC^2 = AB^2 + BC^2. Plugging in the values given, AC^2 = 40^2 + 30^2 = 2500, so AC = sqrt(2500) = 50. Since the diagonals bisect each other, we can conclude that AE = 50/2 = 25. Now, we can use the properties of similar triangles to find the length of BE. Triangle ABE is similar to triangle ACE, so we can set up the following proportion: AB/AC = BE/AE. Plugging in the values, 40/50 = BE/25. Solving for BE, we get BE = (40/50) * 25 = 20.
To find the measure of BE in rectangle ABCD, we use the Pythagorean Theorem. As the diagonals bisect each other, BE is half the length of diagonal AC, resulting in BE = 25.
Explanation:In rectangle ABCD with BC=30 and AB=40, the diagonals bisect each other. That means E, the point of intersection, is the midpoint of both diagonals AC and BD. Since ABCD is a rectangle, its diagonals are equal, and AC = BD. To find BD, we use the Pythagorean Theorem in ΔABC:
AC² = AB² + BC²AC² = 40² + 30²AC² = 1600 + 900AC² = 2500AC = √2500AC = 50Since E is the midpoint of BD, BE is half the length of BD:
BE = BD/2BE = AC/2 (because AC = BD)BE = 50/2BE = 25
Varia is studying abroad in Europe. She is required to pay $3500USD per year to the university, however she must pay in euros. How many euros can Varia expect to pay per month to the university? (Round to the nearest whole euro.)
0.7306 euros = 1 US dollar
A bird flies from the bottom of a canyon that is 59 and three fifths
feet below sea level to a nest that is 609 and one fifth
feet above sea level. What is the difference in elevation between the bottom of the canyon and the bird's nest?
solve x 2 −18x+81=0 for x
The triangular region shows the number of possible raisins, x, and number of possible chocolate chips, y, a baker can use in a recipe.
Which combination of raisins and chocolate chips can the baker use?
A. 28 raisins and 20 chocolate chips
B. 33 raisins and 25 chocolate chips
C. 45 raisins and 10 chocolate chips
D. 55 raisins and 12 chocolate chips
we will proceed to verify each of the cases to determine the solution of the problem
we know that
The combination of raisins and chocolate chips that can the baker use, must be inside the triangular region
so
we're graphing each of the cases
Let
x--------> the number of possible raisins
y--------> the number of possible chocolate chips
case A
Let
[tex]A(28,20)[/tex]
using a graphing tool
see the attached figure
The point [tex]A(28,20)[/tex] is include inside the triangular region
therefore
The case A is a solution
case B
Let
[tex]B(33,25)[/tex]
using a graphing tool
see the attached figure
The point [tex]B(33,25)[/tex] is not include inside the triangular region
therefore
The case B is not a solution
case C
Let
[tex]C(45,10)[/tex]
using a graphing tool
see the attached figure
The point [tex]C(45,10)[/tex] is not include inside the triangular region
therefore
The case C is not a solution
case D
Let
[tex]D(55,12)[/tex]
using a graphing tool
see the attached figure
The point [tex]D(55,12)[/tex] is not include inside the triangular region
therefore
The case D is not a solution
the answer is the option A
[tex](28,20)[/tex]
A. 3x^2+2x-7
B. 3x^2+2x-3
C. 3x^2+8x+3
D. 5x^2+8x-7
Write a linear equation in slope intercept form to model a tree 4 feet tall the grows 3 inches per year
The linear equation that models the growth of a tree that is 4 feet tall and grows 3 inches per year is y = 3x + 48.
Explanation:The problem involves writing a linear equation in slope-intercept form to model the growth of a tree. The slope-intercept form is given by the equation y = mx + b, where 'm' represents the slope or the rate of change, and 'b' is the y-intercept or the initial value.
In the context of this problem, the tree's initial height is 4 feet and it grows at a rate of 3 inches per year. However, the units of the initial height and the yearly growth rate are different, so we need to convert 4 feet into inches. As 1 foot is equivalent to 12 inches, the initial height of the tree is 4 * 12 = 48 inches.
Therefore, the slope or rate of change 'm' is 3 inches per year and the y-intercept or initial value 'b' is 48 inches. Substituting these values into y = mx + b gives us the linear equation representing the growth of the tree: y = 3x + 48.
Learn more about Linear Equations here:https://brainly.com/question/32634451
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Selina claims single having one exemption. Her state tax deduction is 21% of her federal tax contribution. Calculate the amount of state tax Selina owes if her gross pay for two weeks is $840. The following federal tax table is for biweekly earnings of a single person.
Answer:
$16.80
Step-by-step explanation:
All real numbers that are greater than or equal to -1.5 and less than 9.2
A real estate agent sells a house for $92,000. She receives a 6% commission on the sale of the house. How much did she earn on the sale? Round your answer to the nearest cent. Enter your answer in the box. $
The area of a square is found by squaring one of its sides, (A = s 2). A certain square has an area of z 2 + 18z + 81. Factor the trinomial to find a side of the square.
A.) side = z + 27
B.) side = z + 18
C.) side = z + 9
Answer:
C.) side = z + 9
Find P(Not a 2).
X 1 2 3 4
P(X) 0.30 0.40 0.15 0.15
A. 0.70
B. 0.60
C. 0.30
D. 0.40
Answer:
Option B is correct.
Step-by-step explanation:
We have been given the different probabilities at different point we need to find the probability of not of 2
We know that sum of probabilities is 1
[tex]P(a_2)+P(not a_2)=1[/tex]
Here, the probability that is P(a 2)=0.40
Hence, the required probability is 1-0.40=0.60
Therefore, option B is correct.
What are the sine, cosine, and tangent of Θ = 3 pi over 4 radians?
Which graph represents a linear function?
A coordinate plane is shown with a curved line that starts in the bottom left area of the graph, passing through the point -1 comma -5, curving to pass through the y-axis at 2.5, then curving up again to pass through the point 1 comma 5
A coordinate plane is shown with a line passing through the x-axis at negative 2 and the y-axis at negative 6.
A coordinate plane is shown with a parabola graphed in an upward U shape. The bottom of the parabola touches the y-axis at -10 and curves upward on both sides
A coordinate plane is shown with a wavy line that sits on the x-axis and moves across the graph in even up and down waves.
Answer:
2nd Option is correct.
Step-by-step explanation:
We are given with four statements which describes a graph of a function.
We need to find the graph which represents the linear function.
Linear Function:
Linear functions are the functions whose graph are straight lines.
A linear function has the following form
y = f(x) = a + bx
A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.
So, In graphs of 1st , 3rd and 4th Options are curves not a straight line.
Only 2nd Option has graph which is straight as mentioned in it that a line passing through two points.
Therefore, 2nd Option is correct.
if susie is fourteen, what was her age x years ago?
Which size is better in buying detergent: 32 fl oz for $1.99
50fl oz for $2.49
HELP! A simplified version of sin2 θ (1 + cot2 θ) = 1 is ...?
What's the numerator for the following rational expression? 8/y+3/y=?/y
What is the x-intercept of the line with this equation −2r+1/2 y=18?
Enter your answer in the box.
( ,0
state the degree x^3y^2+7x^2y^5-3xy^8
Answer:9
Step-by-step explanation:
The area of one face of a cube is 7 cm2. what is its surface area?
Nicole deposits $2,136 in a savings account paying 5.36% interest. To the nearest dollar, how much money does Nicole have in total after nine years? a. $213 b. $1,030 c. $1,272 d. $3,166
we know that
The simple interest formula is equal to
[tex]A=P(1+rt)[/tex]
where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
in this problem we have
[tex]t=9\ years\\ P=\$2,136\\ A=?\\r=0.0536[/tex]
substitute in the formula above
[tex]A=2,136(1+0.0536*9)[/tex]
[tex]A=2,136(1.4824)[/tex]
[tex]A=\$3,166.41[/tex]
Round to the nearest dollar
[tex]A=\$3,166[/tex]
therefore
the answer is the option D
[tex]\$3,166[/tex]
Write a justification for each step.
Find the greatest common factor of the following monomials.
21g^3h 45g^3h^2
The table lists the values for two parameters, x and y, of an experiment. What is the estimated value of x for y = 0.049?
a) 20.4
b) 21.4
c) 23.7
d) 24.7
yep 20.4 is the correct answer for plato!!
An elliptical track has a major axis that is 80 yards long and a minor axis that is 72 yards long. Find an equation for the track if its center is (0, 0) and the major axis is the x-axis.
Choices are in the attachment
Answer:
So basically it’s D
Step-by-step explanation:
:)
Explain a thought process you might use when trying to find the additive or multiplicative inverse of a number or expression. ...?