Answer:
After x/450 minutes
Step-by-step explanation:
The speed uphill is 200 ft/min
The speed down hill is 250 ft/min
Lets take the hiking distance top to down hill is x ft
Remember speed=distance/time
Relative speed =200+250=450ft/min
Time to meet=distance/relative speed
Time to meet=x/450 min
Final answer:
To find out how long it will take until you meet, we need to determine the distances each hiker covers and then divide it by their respective rates.
Explanation:
To find out how long it will take until you meet, we need to determine the distances each hiker covers and then divide it by their respective rates.
The first hiker covers a certain distance while hiking uphill, while the second hiker covers a certain distance while hiking downhill. When they meet, their combined distances will add up to the total distance of the trail.
Let's say it takes x minutes for you to meet. The first hiker covers a distance of 200x feet in x minutes, while the second hiker covers a distance of 250x feet in the same x minutes. When they meet, their combined distances will equal the length of the trail, so we can set up the equation: 200x + 250x = total distance. Solving for x will give us the time it takes for you to meet.
Pleasr help me ASAP! Thank you^-^ If w = 10 units, x = 5 units, and y = 6 units, what is the surface area of the figure? Round your answer to the nearest tenth.
Please look at picture
A: 656.2 units²
B: 400 units²
C:256.2 units²
D:456.2 units²
Answer:
D:456.2 units²
Step-by-step explanation:
Step 1: Area of all 4 sides of cuboid
Area of rectangle = 4 x length x breadth
Area of rectangle = 4 x 10 x 5
Area of rectangle = 200
Step 2: Calculate bottom of cuboid
Area = length x breadth
Area = 10 x 10
Area = 100
Step 3: Calculate slant height of pyramid
c² = a² + b²
c² is hypotenuse
a² is the base which is half of w (10/2 = 5)
b² is y which is 6
c² = a² + b²
c² = 5² + 6²
c = √61
Step 4: Calculate area of one face of the pyramid
Area of triangle = 1/2 x base x height
Area of triangle = 1/2 x 10 x √61
Area of triangle = 5√61
Step 5: Calculate areas of all 4 faces of the pyramid
4 x 5√61 = 20√61
Step 6: Calculate the total surface area
Total surface area = 200 + 100 + 20√61
Total surface area = 456.2
The surface area of the figure is 456.2 units²
Option D is correct.
!!
Over what interval is the quadratic function decreasing?
I ONLT HAVE 5 min left please answer
Answer:
See below.
Step-by-step explanation:
It decreases in the interval where x > -2.
In interval notation this is (-2, ∞).
The quadratic function decreasing at x>-2.
What is a quadratic function?A quadratic polynomial is a polynomial of degree two in one or more variables.
Given is a graph,
It decreases in the interval where x > -2.
In interval notation, this is (-2, ∞).
Hence, the quadratic function decreasing at x>-2.
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Rahul simplified an expression. His work is shown below.
7(8.5-1.5)+8divided by2
Step 1. 7(7)+8 divided by 2
Step 2. 49+8 divided 2
Step 3. 57 divided by 2
Step 4. 28.5
Where did Rahul make his first mistake?
Answer: Step 3
Step-by-step explanation:
He should have divided 8 by 2 and then added it to 49.
In the third step, Rahul made his first mistake. Simplification is to be done using the BODMAS rule.
What is simplification?Making anything easier to accomplish or comprehend, as well as making it less difficult, is the definition of simplification.
Rahul simplified an expression. His work is shown below.
The expression is given below.
7 (8.5 - 1.5) + 8 / 2
Step 1. 7 (7) + 8 / 2
Step 2. 49 + 8 / 2
Step 3. 49 + 4
Step 4. 53
In the third step, Rahul made his first mistake.
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Use the compound interest formula A =P(1 + r) t and the given information to solve for r.
A =$2300, P =$1600, t =6 r =?
Answer:
Rounding to nearest hundredths gives us r=0.06.
So r is about 6%.
Step-by-step explanation:
So we are given:
[tex]A=P(1+r)^t[/tex]
where
[tex]A=2300[/tex]
[tex]P=1600[/tex]
[tex]t=6[/tex].
[tex]A=P(1+r)^t[/tex]
[tex]2300=1600(1+r)^6[/tex]
Divide both sides by 1600:
[tex]\frac{2300}{1600}=(1+r)^6[/tex]
Simplify:
[tex]\frac{23}{16}=(1+r)^6[/tex]
Take the 6th root of both sides:
[tex]\sqrt[6]{\frac{23}{16}}=1+r[/tex]
Subtract 1 on both sides:
[tex]\sqrt[6]{\frac{23}{16}}-1=r[/tex]
So the exact solution is [tex]r=\sqrt[6]{\frac{23}{16}}-1[/tex]
Most likely we are asked to round to a certain place value.
I'm going to put my value for r into my calculator.
r=0.062350864
Rounding to nearest hundredths gives us r=0.06.
Find the quotient. 9-6i/5+3i
Answer:
see explanation
Step-by-step explanation:
We require to rationalise the denominator by multiplying the numerator and denominator by the complex conjugate of the denominator.
The conjugate of 5 + 3i is 5 - 3i
noting that i² = - 1, hence
[tex]\frac{(9-6i)(5-3i)}{(5+3i)(5-3i)}[/tex] ← expand factors
= [tex]\frac{45-57i+18i^2}{25-9i^2}[/tex]
= [tex]\frac{45-57i-18}{25+9}[/tex]
= [tex]\frac{27-57i}{34}[/tex]
= [tex]\frac{27}{34}[/tex] - [tex]\frac{57}{34}[/tex] i ← quotient
Final answer:
To find the quotient of 9-6i and 5+3i, multiply both the numerator and the denominator by the conjugate of the denominator 5-3i. Simplify by using the distributive property and knowing that i^2 equals -1. The final quotient is 27/34 - 57i/34.
Explanation:
To find the quotient of the complex numbers 9-6i divided by 5+3i, we must multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 5+3i is 5-3i. So, the process is as follows:
Multiply the numerator and the denominator by the conjugate of the denominator: (9-Thus, the quotient is 27/34 - 57i/34 or approximately 0.7941 - 1.6765i.
Can I get some help with this question? Solve 3(x + 2) > x.
Answer:
x > -3
Step-by-step explanation:
We are given the following inequality that we are to solve:
[tex]3(x+2)>x[/tex]
Applying the distributive property of multiplication on the left side of the inequality to get:
[tex] 3 x + 6 > x [/tex]
Rearranging the inequality:
[tex] 3 x - x > - 6 [/tex]
[tex] 2 x > - 6 [/tex]
[tex] x > \frac { - 6 } { 2 } [/tex]
x > -3
Answer:
[tex]\large\boxed{x>-3\to\{x\ |\ x>-3\}\to x\in(-3,\ \infty)}[/tex]
Step-by-step explanation:
[tex]3(x+2)>x\qquad\text{use the distributive property:}\ a(b+c)=ab+ac\\\\(3)(x)+(3)(2)>x\\\\3x+6>x\qquad\text{subtract 6 from both sides}\\\\3x+6-6>x-6\\\\3x>x-6\qquad\text{subtract x from both sides}\\\\3x-x>x-x-6\\\\2x>-6\qquad\text{divide both sides by 2}\\\\\dfrac{2x}{2}>\dfrac{-6}{2}\\\\x>-3[/tex]
Ruben sole t-shirts and hats at a festival his total profit was $168 he made a $6 profit for each t-shirt he sold the also made a profit of $54 from selling hats how many t-shirts did he sell
Answer:
19 t-shirts
Step-by-step explanation:
x= number of t-shirts
6x+54=168; now subtract 54 from both sides of the equation
6x=114; now divide both sides by 6 to isolate the "x"
x=19
Which shows the zeros of f(x) = x^2 - x - 2?
a. 1 and 2
b. -1 and -2
c. 1 and -2
d. 2 and -1
Answer:
d. 2 and -1Step-by-step explanation:
[tex]f(x)=x^2-x-2\\\\\text{The zeros are for}\ f(x)=0:\\\\x^2-x-2=0\\\\x^2-2x+x-2=0\\\\x(x-2)+1(x-2)=0\\\\(x-2)(x+1)=0\iff x-2=0\ \vee\ x+1=0\\\\x-2=0\qquad\text{add 2 to both sides}\\x=2\\\\x+1=0\qquad\text{subtract 1 from both sides}\\x=-1[/tex]
A rectangle has perimeter, P, length, land width, w. Which of the following represents lin
terms of Pand w?
Answer:
[tex]l=\frac{P}{2}-w[/tex].
Step-by-step explanation:
The perimeter of a rectangle is the sum of it's side lengths.
A rectangle has 4 sides where it's opposite sides are congruent.
So if one side has measurement w, then there is another side that has measurement w.
If there is one side that has measurement l, then there is another side that has measurement l.
So if you add w+w+l+l you get 2w+2l.
They are giving us that the perimeter is P, so P=2w+2l.
we are being asking to solve for l.
P=2w+2l
First step: Isolate term that contains the l, so get 2l by itself first.
We are going to subtract 2w on both sides giving us:
P-2w=2l
2l=P-2w
Now that we have 2l by itself it is time to perform the last step in getting l by itself.
Second step: Divide both sides by 2.
This gives us:
l=(P-2w)/2
You may separate the fraction like so:
[tex]l=\frac{P-2w}{2}=\frac{P}{2}-\frac{2w}{2}=\frac{P}{2}-w[/tex].
I don't know your options but I have solve for l in terms of P and w
and got [tex]l=\frac{P}{2}-w[/tex].
Please let me know if you have further questions with this problem.
HELP ME!! Please shoe me steps too please
Answer:
C
Step-by-step explanation:
The equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b) a point on the line
y - 5 = - 3(x + 2) ← is in point- slope form
with slope m = - 3
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-3}[/tex] = [tex]\frac{1}{3}[/tex]
and (a, b) = (6, - 1), hence
y - (- 1) = [tex]\frac{1}{3}[/tex](x - 6), that is
y + 1 = [tex]\frac{1}{3}[/tex](x - 6) → C
Simplify
9 to the 2
over 9 to the 7
Answer:
= 1/59049 ....
Step-by-step explanation:
9 to the 2 = 9^2 = 9*9 = 81
9 to the 7 = 9^7= 9*9*9*9*9*9*9 = 4782969
Now simplify the values by table of 9
=81/ 4782969
=9/531441
=1/59049 ....
Step-by-step explanation:
9 to the 2ND power is 81
9 to the 7th power is 4,782,969
81 divided by 81 is 1
4,782,969 divided by 81 is 59,049
so the final answer is 1
59,049
through: (4,-1), perpendicular to y=x+2
Answer:
y = - x + 3
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = x + 2 ← is in slope- intercept form
with slope m = 1
Given a line with slope m then the slope of a perpendicular line is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{1}[/tex] = - 1, hence
y = - x + c ← is the partial equation of the perpendicular line
To find c substitute (4, - 1) into the partial equation
- 1 = - 4 + c ⇒ c = - 1 + 4 = 3
y = - x + 3 ← equation of perpendicular line
Solve for x.
A. 11
B. 17
C. 15
D. 19
The full length of one line times the length of the line outside the circle is equal the the other line.
(7+x)*7 = (13 +8) * 8
Simplify:
7x +49 = 21 * 8
7x +49 = 168
Subtract 49 from each side:
7x = 119
Divide both sides by 7:
x = 17
The answer is B. 17
The value of x for the given circle will be 17 so option (B) will be correct.
What is a circle?A circle is a geometrical figure which becomes by plotting a point around a fixed point by keeping a constant distance.
In our daily life, we always see circle objects for example our bike wheel.
The longest line which can be drawn inside the circle will be the diameter.
Area of circle = πr² and the perimeter of circle = 2πr where r is the radius of the circle.
By theorem in circle
( 13 + 8) × 8 = ( x + 7) × 7
21 × 8 = ( x + 7) × 7
x + 7 = 3 × 8
x = 24 - 7
x = 17
Hence, The value of x for the given circle will be 17.
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The ratio of the height of two similar cylinders is 4:1. What is the ratio of their volumes?
Answer:
64 : 1
Step-by-step explanation:
Given 2 similar cylinders with linear ratio = a : b then
volume ratio = a³ : b³
Here the height ratio = 4 : 1, hence
volume ratio = 4³ : 1³ = 64 : 1
A river is flowing downstream at rate of 2 km/h. Murray can swim at a rate of 3 km/h. Murray jumps in and swims downstream for a certain distance then turns around and swims upstream back to the start. In total it takes 30 minutes. How far did Murray swim downstream?
Answer:
Murray can swim 3/4 km downstream
Step-by-step explanation:
Since the current adds to his speed and
then subtracts the same amount, you can
disregard the current
d=r*t
Convert 15 min to hrs ( time to swim downstream )
15(1/60)=1/4 hours
d=3(1/4)
d=3/4 km downstream.
Hence Murray can swim 3/4 km downstream....
Final answer:
Murray swam 0.417 km downstream before turning around and swimming back upstream. The calculation involves solving a simple algebraic equation using Murray's swimming speed and the river's flow rate, within the total time frame of 30 minutes.
Explanation:
The student is asking about a problem involving rates and time, which is a common topic in algebra and physics. In this scenario, Murray is able to swim at a speed of 3 km/h in still water, and the river flow adds an extra 2 km/h when swimming downstream, making his effective downstream speed 5 km/h. When swimming upstream, Murray has to work against the river flow, reducing his effective speed to 1 km/h (3 km/h - 2 km/h). Given that the total time spent swimming is 30 minutes (0.5 hours), we need to determine the distance Murray swam downstream before turning back.
Let the distance Murray swam downstream be d kilometers. The time to swim downstream at 5 km/h is d/5 hours, and the time to swim back upstream at 1 km/h is d hours. The sum of these times equals the total time Murray was swimming:
d/5 + d = 0.5
By solving the equation, we find that d = 0.417 km. Therefore, Murray swam 0.417 kilometers downstream before turning around and swimming back upstream.
Chaos
Question 7
(02 04 MC)
por
A car manufacturer is reducing the number of incidents with the transmission by issuing a voluntary recall. During week 10 of the recall, the manufacturer fixed 200 cars. In week 15, the manufacturer fixed 175 cars. Assume
that the reduction in the number of cars each week is linear Write an equation in function form to show the number of cars seen each week by the mechanic.
f(x) = 5x + 250
f(x) = -5x+250
f(X) = 10x+200
f(x) = -10x+200
Answer:
The correct option is B) [tex]f(x)=-5x+250[/tex].
Step-by-step explanation:
Consider the provided information.
let x represents the number of weeks and y represents the fixed cars.
During week 10 of the recall, the manufacturer fixed 200 cars.
Thus, the ordered pair can be made with the help of the above data is: (10,200).
In week 15, the manufacturer fixed 175 cars.
Thus, the ordered pair can be made with the help of above data is: (15,175)
Now use two point slope formula: [tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
Substitute the [tex](x_1,y_1)=(10,200)\ \text{and}\ (x_2,y_2)=(15,175)[/tex] in the above formula.
[tex]y-200=\frac{175-200}{15-10}(x-10)[/tex]
[tex]y-200=\frac{-25}{5}(x-10)[/tex]
[tex]y-200=-5(x-10)[/tex]
[tex]y-200=-5x+50[/tex]
[tex]y=-5x+250[/tex]
Which also can be written as:
[tex]f(x)=-5x+250[/tex]
Hence, the correct option is B) [tex]f(x)=-5x+250[/tex].
5 MINUTE TIME LIMIT LEFT Consider the polynomial expression below.
[tex]x^{3} -3x^{2} +81x-243[/tex]
Rewrite the polynomial in the form (x - d)(x - e)(x + f), where d is a real number and e and f are complex numbers of the form bi.
[tex]x^3-3x^2+81x-243=\\x^2(x-3)+81(x-3)=\\(x^2+81)(x-3)=\\(x-3)(x-9i)(x+9i)[/tex]
If 47400 dollars is invested at an interest rate of 7 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent.
(a) Annual:______ $
(b) Semiannual:_____ $
(c) Monthly:______ $
(d) Daily:_______ $
Answer:
Part A) Annual [tex]\$66,480.95[/tex]
Part B) Semiannual [tex]\$66,862.38[/tex]
Part C) Monthly [tex]\$67,195.44[/tex]
Part D) Daily [tex]\$67,261.54[/tex]
Step-by-step explanation:
we know that
The compound interest formula is equal to
where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest in decimal
t is Number of Time Periods
n is the number of times interest is compounded per year
Part A)
Annual
in this problem we have
[tex]t=5\ years\\ P=\$47,400\\ r=0.07\\n=1[/tex]
substitute in the formula above
[tex]A=47,400(1+\frac{0.07}{1})^{1*5} \\A=47,400(1.07)^{5}\\A=\$66,480.95[/tex]
Part B)
Semiannual
in this problem we have
[tex]t=5\ years\\ P=\$47,400\\ r=0.07\\n=2[/tex]
substitute in the formula above
[tex]A=47,400(1+\frac{0.07}{2})^{2*5} \\A=47,400(1.035)^{10}\\A=\$66,862.38[/tex]
Part C)
Monthly
in this problem we have
[tex]t=5\ years\\ P=\$47,400\\ r=0.07\\n=12[/tex]
substitute in the formula above
[tex]A=47,400(1+\frac{0.07}{12})^{12*5}\\A=47,400(1.0058)^{60}\\A=\$67,195.44[/tex]
Part D)
Daily
in this problem we have
[tex]t=5\ years\\ P=\$47,400\\ r=0.07\\n=365[/tex]
substitute in the formula above
[tex]A=47,400(1+\frac{0.07}{365})^{365*5}\\A=47,400(1.0002)^{1,825}\\A=\$67,261.54[/tex]
What is the solution to this equation?
– 8х + 4 = 36
ОА. x = -5
ОВ. x= -4
Ос. х = 5
O D. x= 4
b) -4
- 8x +4 =36
First, we subtract 4 from 36 to get 32
36-4=32
-8x=32
Since a negative times a negative equals a positive, then the answer has to be negative because 36 is positive.
32 divided by 8 = 4
B) -4
Answer:
The answer is B, x=-4
Step-by-step explanation:
-8x + 4 = 36
-8x - 4 = 36 - 4
-8x = 32
-8x/-8 = 32/-8
x = -4
Find x (x+4)÷3 = (x+8)÷4
Answer:
x=8
Step-by-step explanation:
(x+4) = (x+8)
------ -------
3 4
Multiply each side by 12 to get rid of the fraction
(x+4) = (x+8)
12 *------ ------- *12
3 4
4*(x+4) = (x+8)*3
Distribute
4x+16 = 3x+24
Subtract 3x from each side
4x-3x+16 = 3x-3x+24
x+16 = 24
Subtract 16 from each side
x+16-16 = 24-16
x = 8
When is a rhombus a rectangle?
Answer: D
Step-by-step explanation:
They are rectangles when the sides are 90 degrees. :)
A rhombus is a rectangle when all its angles are right angles.
What is a rectangle?
A rectangle is a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square.
Given is a rhombus.
A rhombus is a rectangle when all its angles are right angles. Due to this, the tilt of the vertical sides of rhombus will become 0 degrees.
Therefore, a rhombus is a rectangle when all its angles are right angles.
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Which of the slope is an equation of a line whose slope is 0?
1): Y=6
2):x=6
3):y=2x
4):x+y=1
Answer:
1) y=6
Step-by-step explanation:
The equation y=mx+b is called slope-intercept because it tells us the slope,m, and y-intercept ,b.
The equation y=a is a horizontal line and goes through a on the y-axis. Horizontal lines have a slope of zero.
The equation x=b is a vertical line and goes through b on the x-axis.
Vertical lines have an undefined slope.
1) y=6 is horizontal so it's slope is 0
2) x=6 is vertical so it's slope is undefined
3) y=2x has slope 2
4) x+y=1 can be put into the form y=mx+b to determine the slope.
Subtract x on both sides:
y=-x+1
The slope is -1.
In the drawing below, line n is a transversal that intersects two parallel lines. What is the measure of Angle 4?
40°
50°
130°
140°
Answer:
140°
Step-by-step explanation:
<3 = 40 because they are vertical angles
<3 +<4 = 180 because they are same side interior angles
40 + <4 = 180
Subtract 40 from each side
40-40 + <4 = 180-40
<4 = 140
Write the equation of the line shown in the graph. Please help.
Answer:
y = 3
Step-by-step explanation:
The y-coordinate of all points on the graph is 3.
The equation is y = 3
Answer:
y = 3
Step-by-step explanation:
The equation of a horizontal line parallel to the x- axis is
y = c
Where c is the value of the y- coordinates the line passes through.
In this case the line passes through points with a y- coordinate of 3, hence
Equation of horizontal line is y = 3
Lynne is hiking. When she stands at the base of the mountain, the horizontal
distance between Lynne and the highest point of the mountain is 588 feet. If
the mountain has an elevation of 2610 feet, what is the angle of elevation
from her starting point to the highest point of the mountain? Round your
answer to the nearest whole number.
Answer:
Angle of elevation of the highest point of the mountain is 77°
Step-by-step explanation:
Lynne is hiking. When she stands at the base of the mountain, the horizontal distance between Lynne and the highest point of the mountain is 588 feet.
Elevation of the mountain is 2610 feet.
We have to calculate the angle of elevation ∠C.
tanC = [tex]\frac{2610}{588}[/tex]
tanC = 4.439
C = [tex]tan^{-1}(4.439)[/tex]
C = 77.30 ≈ 77°
Therefore, angle of elevation of the mountain is 77°
approximately how many cubic feet of water could the tower hold?
I WILL MARK BRAINLIEST
(answer options and full questions is in the image above)
G. 7,500 cubic feet
Explanation:Find the radius of the tower. [tex]r=\frac{d}{2}=\frac{20}{2}=10[/tex]The formula for the volume of a cylinder is [tex]V=\pi r^2 h[/tex]. Substitute the values. [tex]V=\pi * 10^2 * 24[/tex]Find the exponent. [tex]V=\pi * 100 * 24[/tex]Multiply. [tex]V=\pi * 2400[/tex]Estimate by setting [tex]\pi[/tex] equal to 3.14. [tex]V=3.14*2400=7536[/tex]The closest answer choice is G. 7,500 cubic feet.Suzie bought an antique jewelry box for $34.50. It originally cost $8.75 when it was made. Its value has increased $0.75 a year since it was made and it is now 83 years old. If Suzie were to sell it for what it is worth, how much money would she make? (Please show steps)
Answer:
if she would sell it for what it is worth now she would make $36.50
Step-by-step explanation:
0.75 x 83 = 62.5
62.5 + 8.75 = 71
71 is how much it is worth now
71 - 34.5 = 36.5
36.50 is how much she would make
Answer:
$36,50
Step-by-step explanation:
That is correct to me. You do those exact steps to arrive at that answer.
Which lists all of the x-intercepts of the graphed function?
(0, 6)
(1, 0) and (2, 0)
(1, 0), (2, 0), and (–3, 0)
(1, 0), (2, 0), (–3, 0), and (0, 6)
Answer:
B
Step-by-step explanation:
The x- intercepts are the points on the x- axis where the graph crosses.
That is (- 3, 0), (1, 0) and (2, 0) → set B
Answer:
D. (1,0), (2,0), (-3,0), and (0,6)
Step-by-step explanation:
i only need the answer to part b please
Answer:
3 tyres
Step-by-step explanation:
56/4=14
14*3=42
Answer:
3 tyres he will fill.....
Find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively.
Answer:
[tex]a_n=7 \cdot (-3)^{n-1}[/tex]
Step-by-step explanation:
The explicit form for a geometric sequence is [tex]a_n=a_1 \cdot r^{n-1}[/tex] where [tex]a_1[/tex] is the first term and [tex]r[/tex] is the common ratio.
We have the following given:
[tex]a_2=-21[/tex]
[tex]a_5=567[/tex].
We also know that [tex]a_2=a_1 \cdot r[/tex] while [tex]a_5=a_1 \cdot r_4[/tex].
So if we do 5th term divided by second term we get:
[tex]\frac{a_1 \cdot r_4}{a_1 \cdot r}=\frac{567}{-21}[/tex]
Simplifying both sides:
[tex]r^3=-27[/tex]
Cube root both sides:
[tex]r=-3[/tex]
The common ratio, r, is -3.
Now we need to find the first term.
That shouldn't be too hard here since we know the second term which is -21.
We know that first term times the common ratio will give us the second term.
So we are solving the equation:
[tex]a_1 \cdot r=a_2[/tex].
[tex]a_1 \cdot (-3)=-21[/tex]
Dividing both sides by -3 gives us [tex]a_1=7[/tex].
So the equation is in it's explicit form is:
[tex]a_n=7 \cdot (-3)^{n-1}[/tex]
Check it!
Plugging in 2 should gives us a result of -21.
[tex]a_2=7 \cdot (-3)^{2-1}[/tex]
[tex]a_2=7 \cdot (-3)^1[/tex]
[tex]a_2=7 \cdot (-3)[/tex]
[tex]a_2=-21[/tex]
That checks out!
Plugging in 5 should give us a result of 567.
[tex]a_5=7 \cdot (-3)^{5-1}[/tex]
[tex]a_5=7 \cdot (-3)^4[/tex]
[tex]a_5=7 \cdot 81[/tex]
[tex]a_5=567[/tex]
The checks out!
Our equation works!
Final answer:
To find the nth term formula of a geometric sequence with given terms, divide one term by the other to find the common ratio, and then solve for the first term. For this sequence, the nth term is [tex]a_{n}= 7 (-3)^{n-1}[/tex].
Explanation:
To find an equation for the nth term of a geometric sequence where the second and fifth terms are -21 and 567, respectively, we must determine the common ratio (r) and the first term (a1) of the sequence. For a geometric sequence, the nth term is given by the formula [tex]a_{n}= a_{1} (r)^{n-1}[/tex].
Since the second term a2 is -21 and the fifth term a5 is 567, we can set up the following equations using the geometric sequence formula:
[tex]a_{2}[/tex] = [tex]a_{1}[/tex] x r = -21
[tex]a_{5}[/tex] = [tex]a_{1}[/tex] x [tex]r_{4}[/tex] = 567
Dividing the second equation by the first gives us:
[tex]r_{3}[/tex] = 567 / -21 = -27
Thus, the common ratio r is -3. Now using [tex]a_{2} =a_{1} r[/tex] , we find that [tex]a_{1}[/tex] = -21 / (-3) = 7. Therefore, the nth term of the sequence is:
[tex]a_{n}= 7 (-3)^{n-1}[/tex]