Therefore, it will take approximately 13.47 years for $2000.00 to double at an annual rate of 5.25% compounded monthly.
What is percent?Percent is a way of expressing a number as a fraction of 100. The symbol for percent is "%". Percentages are used in many different contexts, such as finance, economics, statistics, and everyday life. Percentages can also be used to express change or growth, such as an increase or decrease in the value of something over time.
Here,
The formula for the accumulated amount (A) when interest is compounded n times per year at an annual interest rate of r, for t years, is:
[tex]A = P(1 +\frac{r}{n})^{nt}[/tex]
where P is the principal amount (initial investment).
To find approximately how long it will take $2000.00 to double at an annual rate of 5.25% compounded monthly, we need to solve for t in the above formula.
Let P = $2000.00, r = 0.0525 (5.25% expressed as a decimal), and n = 12 (monthly compounding).
Then, we have:
[tex]2P = P(1 +\frac{r}{n})^{nt}[/tex]
Dividing both sides by P, we get:
[tex]2= (1 +\frac{r}{n})^{nt}[/tex]
Taking the natural logarithm of both sides, we get:
[tex]ln(2) =ln(1 +\frac{r}{n})^{nt}[/tex]
Using the properties of logarithms, we can simplify this expression as:
[tex]ln(2) = n*t * ln(1 + r/n)[/tex]
Dividing both sides by n*ln(1 + r/n), we get:
[tex]t = ln(2) / (n * ln(1 + r/n))[/tex]
Plugging in the values for r and n, we get:
[tex]t = ln(2) / (12 * ln(1 + 0.0525/12))[/tex]
Solving this expression on a calculator, we get:
t ≈ 13.47 years
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PLEASE HELP ASAP DUE IN A HOUR
Answer:
51
Step-by-step explanation:
3x+4y^2
3(5)+4(3)^2
3(5)+4(9)
15+36
51
Answer: 51
Step-by-step explanation:
in which place is the first digit of the quotient: 3,587 ÷ 18
Answer: The quotient is 199.277777778, the answer is the hundreds place
Step-by-step explanation:
199.277777778, the 1 in 199 is in the hundreds place :)
Question 4
What are the coordinates of point C on the directed segment
from A(-8,4) to B(10,-2) that partitions the segment such that
AC:CB is 2:1?
We may conclude after answering the presented question that Therefore, the coordinates of point C are (-6, 0).
what are coordinate ?In mathematics, a coordinate is a number or combination of numbers that represents the location of a point or object in space. Coordinates are frequently used to define a point or object's position in a certain coordinate system, such as the Cartesian or polar coordinate systems. In the Cartesian coordinate system, a point is found by its distance from the origin along the x-axis and its distance from the origin along the y-axis. These distances are represented by two integers, the x-coordinate and the y-coordinate. The point's location in the plane is specified by the two coordinates.
Using the distance formula, we can find the distances AC and CB:
[tex]AC = \sqrt((x - (-8))^2 + (y - 4)^2)\\CB = \sqrt((10 - x)^2 + (-2 - y)^2)\\[/tex]
Since AC:CB = 2:1, we have:
[tex]AC/CB = 2/1\sqrt((x - (-8))^2 + (y - 4)^2) / \sqrt((10 - x)^2 + (-2 - y)^2) = 2/1\\(x - (-8))^2 + (y - 4)^2 / ((10 - x)^2 + (-2 - y)^2) = 4\\(x - (-8))^2 + (y - 4)^2 = 4((10 - x)^2 + (-2 - y)^2)\\5x^2 + 5y^2 - 116x + 20y + 340 = 0\\(x - (-8))^2 + (y - 4)^2 = r^2\\(x - (-8))^2 + (y - 4)^2 / ((10 - x)^2 + (-2 - y)^2) = 4\\[/tex]
where r is the radius of the circle.
Solving this system of equations, we get:
x = -6
y = 0
Therefore, the coordinates of point C are (-6, 0).
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an athlete ran 200m race in 25 seconds. how fast did he run in meters per second
Answer:
To calculate the speed of the athlete in meters per second (m/s), we can use the formula:
Speed = Distance / Time
Here, the distance is 200 meters and the time is 25 seconds. Substituting these values into the formula, we get:
Speed = 200 meters / 25 seconds
Simplifying, we get:
Speed = 8 meters/second
Therefore, the athlete ran at a speed of 8 meters per second.
The perimeter of a rectangle is 34 units. Its width is 6.5 units.
Write an equation to determine the length of the rectangle
Given:
P = 34 [Perimeter]
W = 6.5 {let's use W and L to make it easy to read]
Also known:
P = W + L + W + L = 2(W+L)
Fill in values:
34 = 2(6.5+L) [NOTE: This is an equation to determine the length (L) of the rectangle."]
17 = 6.5 + L [NOTE: This is a simplified form of the above equation.]
10.5 = L [NOTE: This is the solution (the value of L)]
9. If the figure below is made of cubes with 2 cm side lengths, what is its volume? 14 cu. cm. 42 cu. cm. 144 cu.cm 280 cu. cm.
is this correct??
Total volume of shape is =144cu.cm
Formula for Cube VolumeBy knowing the length of the cube's edges, we can quickly determine its volume (V). Assume that "a" is the cube's edge length. The product of length, height, and breadth will then be the V. The cube's volume formula is as follows:
Cube Volume = Length× Width× Height
Volume equals= a× a× a =a³
where "a" denotes the cube's side or edges' length.
GivenLength of each cube = 2cm
Volume of each cube=a³
=2³
=8cm³
Total number of cubes=18
Total volume of shape=18×8
=144 cu.cm
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For a given recipe 8 cups of flour are mixed with 12 cups of sugar how many cups of flour should be used if 27 cups of sugar are use
Therefore, if we are using 27 cups of sugar, we also need 18 cups of wheat.
What does an arithmetic ratio mean?An ordered combination of integers a and b, rendered as a / b, is a ratio if b is not equal to 0. A percentage is an expression that sets two numbers at the same value. For instance, you could put the percentage as follows: 1: 3 if it's 1 male and 3 girls. (for every one boy there are 3 girls)
If 8 cups of flour are mixed with 12 cups of sugar, then the ratio of flour to sugar is:
8 : 12
Simplifying this ratio by dividing both numbers by 4, we get:
2 : 3
This means that for every 2 cups of flour, we need 3 cups of sugar.
If we are using 27 cups of sugar, we can set up a proportion to find out how much flour we need:
2/3 = x/27
Multiplying both sides by 27:
2 × 27 / 3 = x
Simplifying:
x = 18
Therefore, we need 18 cups of flour if we are using 27 cups of sugar.
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how do you do this mathematics?
Answer:
What Mathematics
Step-by-step explanation:
As shown in the figure below, Kaitlin is standing 93 feet from the base of a leaning tree. The tree is growing at an angle of
88° with respect to the ground. The angle of elevation from where Kaitlin is standing to the top of the tree is 35°. Find the
length, x, of the tree. Round your answer to the nearest tenth of a foot.
35°
-93 ft-
H
88°
feet
X
5
Answer:The measure of Ф = 75.7° :)
Step-by-step explanation:the measure of Ф = 75.7°,We will use the rule sin to find the measure of Ф
At first we will find the angle opposite to the side of length 31
∵ 180° - 73° = 107°
hope this helps:)(:
If the diameter of a circle is
30
30 centimeters, what is the radius of the circle?
Answer:
15 centimeters
Step-by-step explanation:
radius is half of the circles diameter
June is driving from Brookline, Massachusetts, to Brooklyn, New York. The cities are 200 miles apart. Pretend June lives in a world in which she encounters no traffic jams and can drive at a constant speed of 50 mph the entire way. The distance June has traveled, c, is a function of her driving time, t Write a formula that describes this function.
This function's definition is given by the formula [tex]c = 50t + 200[/tex].
What pace is it moving at?Consider how many individuals and gadgets will be utilizing the network at once as well as how it will be utilized before choosing the "optimal" speed for your home. As a general rule, anything that can connect many devices simultaneously and is above 200 Mbps is regarded as "fast" internet.
What would you define speed as?Speed, which is a scalar number, is the "speed at which an item is moving." The pace at which an item travels a distance may be referred of as its speed. A fast-moving item travels at a great velocity and completes a significant distance in a brief period of time.
[tex]y = mx + b[/tex]
Where [tex]y,x[/tex] are variables, m is the rate of change and [tex]b[/tex] is the y intercept.
Let [tex]c[/tex] represent the distance June has traveled after time [tex]t[/tex].
Since the cities are [tex]200[/tex]miles apart, hence [tex]b = 200[/tex]. Also the speed is [tex]50[/tex]mph, hence [tex]m = 50[/tex]:
[tex]c = 50t + 200[/tex]
The formula that describes this function is [tex]c = 50t + 200[/tex]
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Solve the following quadratic by completing the square
f(x) = x^2 + 2x-12
The quadratic equation f(x) = x² + 2x - 12 can be rewritten as f(x) = (x + 1)²- 13 using the completing square method.
What is completing squares method?Completing the square is a strategy used in algebra to solve quadratic equations of the type ax² + bx + c = 0, where a, b, and c are constants and x is the variable. By converting the quadratic equation into a perfect square trinomial, the square roots of both sides of the equation may be used to quickly solve the problem. This is the core notion behind the completion of the square. The quadratic expression becomes a perfect square when we add and subtract a specific number from it to complete the square. Half of the x-term squared coefficient is equal to the sum of the numbers we add and remove.
The given function is f(x) = x² + 2x-12.
Now, group the x terms together:
f(x) = (x² + 2x) - 12
Between the parenthesis, add and remove the square of the x-half-coefficient: term's:
f(x) = (x² + 2x + (2/2)² - (2/2)²) - 12
f(x) = (x + 1)² - 1 - 12
f(x) = (x + 1)² - 13
Hence, the quadratic equation f(x) = x² + 2x - 12 can be rewritten as f(x) = (x + 1)² - 13.
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Your take home pay is $3,210 and your federal witholding tax rate is 16%. What was your gross pay? show the math
Answer:
To find your gross pay, we can use the following formula:
Gross Pay = Take Home Pay / (1 - Federal Withholding Tax Rate)
Plugging in the given values, we get:
Gross Pay = 3,210 / (1 - 0.16)
Simplifying the denominator, we get:
Gross Pay = 3,210 / 0.84
Calculating the division, we get:
Gross Pay = 3,821.43
Therefore, your gross pay was $3,821.43.
Decomposing numbers to add make a new 10 adding.
29+14
29 + 14 = 44 by decomposing numbers to add and make a new 10.
Describe Decomposition of numbers?Decomposition of numbers involves breaking a number down into its constituent parts or smaller numbers that add up to the original number. This can be done in different ways depending on the purpose and context of the problem.
For example, one common way of decomposing a number is by place value. In this case, the number is broken down into its digits, which represent different powers of ten. For instance, the number 562 can be decomposed into 500 + 60 + 2.
Another way of decomposing numbers is by factoring. In this case, the number is expressed as a product of its factors. For example, the number 12 can be decomposed as 2 x 2 x 3.
Decomposing numbers is a useful skill in many areas of mathematics, including arithmetic, algebra, and geometry. It is often used to simplify problems, make computations easier, and facilitate further analysis.
To decompose 29 to make a new 10, we can break it into 9 and 20. Then, to decompose 14 to make a new 10, we can break it into 6 and 8.
So, we have:
29 + 14 = (9 + 20) + (6 + 8)
Now, we can regroup the numbers to make a new 10:
= (9 + 1) + (20 + 6) + 8
= 10 + 26 + 8
= 44
Therefore, 29 + 14 = 44 by decomposing numbers to add and make a new 10.
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100 POINTS + BRAINLIEST!!
Answer:
Taking radius of a semicircle to be 3cm calculate the area using (1/2πr²) and add the area of rectangle calculated by use of formula L×w
Therefore Area=π(1/2×3²)+(12×6)
=76.5π
Answer:
82.27
Step-by-step explanation:
Please see the diagram attached to this solution for measurements
Area of the figure = Area of semi circle + area of rectangle
As per the figure
[tex]l_{rectangle} = 9\\w_{rectangle} = 6\\r_{semi-circle} = 3[/tex]
therefore,
[tex]Area = \dfrac{\pi r^{2}}{2} + l\cdot w[/tex]
substitute values of l, w and r, we get
[tex]Area = 82.27 cm^2[/tex]
Hopefully this answer helped you!!!
What is 17 + 8 x (2.7 ÷ 6) - 3
using
P
E
M
D
A
S
Answer:
17.6
Step-by-step explanation:
2.7÷6= 0.45
0.45×8= 3.6
17+3.6 - 3=17.6
Quelle est l’explication de la puissance de 10
Answer:
Step-by-step explanation:
Lorsque l'exposant (a) est positif, alors la puissance de dix 10a correspond au nombre 1 suivi d'un nombre de zéros correspondant au chiffre a. Quelques exemples : 103 correspond au nombre 1 suivi de 3 zéros donc 103 = 1 000.
Adam is finding 10 less than 708 mentally. He thinks the tens digit and the hundreds digit will change. He gets 698 for his answer. Is Adam's thinking correct? Explain.
Answer:
Adam's thinking is not correct.
When subtracting 10 from 708, we need to borrow 1 from the hundreds digit and add it to the tens digit. This will result in a new hundreds digit of 6 and a new tens digit of 9. The ones digit remains the same. Therefore, the correct answer would be 698.
Adam's answer is also 698, but his reasoning is incorrect. He thinks that both the tens and hundreds digits change, which is not the case. In reality, only the tens digit changes while the hundreds digit decreases by 1 due to the borrowing process.
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Lyla invests $2,500 into a savings account
which earns 5% per year. In 15 years, how
much will Lyla's investment be worth if interest
is compounded semiannually (twice a year)?
Round to the nearest dollar.
Answer:
We can use the formula for compound interest to find the future value of Lyla's investment:
A = P(1 + r/n)^(nt)
where A is the future value, P is the principal (initial investment), r is the annual interest rate as a decimal, n is the number of times interest is compounded per year, and t is the time in years.
Substituting the given values, we get:
A = $2,500(1 + 0.05/2)^(2*15)
Simplifying, we get:
A = $2,500(1.025)^30
Using a calculator, we get:
A ≈ $5,016.35
Therefore, Lyla's investment will be worth approximately $5,016.35 in 15 years if interest is compounded semiannually. Rounded to the nearest dollar, the answer is $5,016.
If the graph of a polynomial function P(x) has -intercepts at x = - 4, x = 0, x * 1 point
= 5, which of the following must be true for P(x)?
• (x + 5) is a factor of the polynomial.
• (x-4) is a factor of the polynomial.
•' The degree of the polynomial is 3.
• The degree of the polynomial is greater than or equal to 3.
(x + 5) is nοt necessarily a factοr οf the pοlynοmial, (x-4) is a factοr οf the pοlynοmial are cοrrect statement.
What is a functiοn ?Functiοn can be define in which it relates an input tο οutput.
If the graph οf a pοlynοmial functiοn P(x) has x-intercepts at x = -4, x = 0, and x = 5, then we knοw that the factοrs οf P(x) are (x + 4), x, and (x - 5). This is because a pοlynοmial has x-intercepts where the value οf P(x) is equal tο zerο, and this οccurs when each factοr is equal tο zerο.
Therefοre, we can cοnclude that (x + 4) and (x - 5) are factοrs οf the pοlynοmial P(x), but x is nοt necessarily a factοr. This is because x is a linear factοr with a zerο intercept, but it cοuld be cancelled οut by anοther factοr in the pοlynοmial.
Thus, the cοrrect statement is:
(x + 5) is nοt necessarily a factοr οf the pοlynοmial.
(x-4) is a factοr οf the pοlynοmial.
The degree οf the pοlynοmial is 3 οr greater since the pοlynοmial has three x-intercepts. Hοwever, we cannοt determine the exact degree οf the pοlynοmial withοut additiοnal infοrmatiοn.
Therefοre, (x + 5) is nοt necessarily a factοr οf the pοlynοmial, (x-4) is a factοr οf the pοlynοmial are cοrrect statement.
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What is the value of x in (x+3)^2=49 show your work
Answer:
x = 4,-10Step-by-step explanation:
(x+3)^2=49
x+3=√(49)
x+3=√(7^2)
x+3=7
x+3-3=7-3
x=4x+3=-√(49)
x+3=-7
x+3-3=-7-3
x=-10x = 4,-10
There is a park near Raphael’s home. To find its area, Raphael took the measurements shown. Select all the true statements about the area of the park.
(a)The park can be decomposed into two parallelograms.
(b)The formula A = bh can be used to find the area of each piece of the park.
(c)The park can be decomposed into a parallelogram, a triangle, and a trapezoid.
(d)The area of the park is 126 m2.
(e)The area of the park is 96 m2.
Answer:
A, C, and D
Step-by-step explanation:
If you split the shape right along the bottom of the first part, you get two parallelograms. If you split it like you split it the first time and then split it along the side of the bottom parallelogram you get a triangle, a parallelogram, and a trapezoid. Then, because the area of a parallelogram is A=bh, you can substitute the numbers in. 5*12=60 and 6*11=66. However, this is not all, remember to add the area together! 60+66=126 so D is also true.
Hope this helps!
Anna has one dollar she buy some candy for $.83 which shows how much change and I should get
Answer:
Anna should receive $0.17 (17 cents) in change
Step-by-step explanation:
Anna has $1, she spends $.83 on candy
To find how much change she will get, we must subtract
1 - 0.83
$0.17
Answer:$0.17
Step-by-step explanation:
A dollar is 100 cents just subtract 83 from 100 to get 17.
100 Points!!! Algebra question, multiple choice. Only looking for an answer to #8. Find the maximum value of f(x,y)=3x+y for the feasible region. Photo attached. Thank you!
Answer:
+4
Step-by-step explanation:
F(x,y) = 3x+y and y <= -2x+ 4 sub in for 'y'
= 3x + (-2x+4)
= x + 4
If you look at the graph for y <= - 2x+4 ( see below)
you will see that the domain (x values ) can only go from 0 to 4 and the max value is +4 ( rememeber too that y is restricted to >= 0 as is x )
a toy rocket is shot vertically into the air from a 9-foot-tall launching pad with an initial velocity of 144 feet per second. Suppose the height of the rocket in feet t seconds after being modeled by the function h(t)=-16² gthg. where v is the initial velocity of the rocket and he is the initial height of the rocket. How long will it take for the rocket to reach its maximum The rocker will reach its maximum height in second(s) launched can be height what s
Answer:
9 is answer you fool......
answer: Click THANKS if you like my answer. have a good day sir/maam #keep safe
First, we need to find the maximum height that the rocket will reach. To do that, we need to find the vertex of the parabolic function h(t) = -16t^2 + 144t + 9. We can use the formula for the vertex of a parabola, which is given by the formula t = -b/2a, where a = -16 and b = 144.
t = -b/2a = -144/(2*(-16)) = 4.5
So the rocket will reach its maximum height after 4.5 seconds.
To find the maximum height, we need to substitute t = 4.5 into the function h(t):
h(4.5) = -16(4.5)^2 + 144(4.5) + 9 = 324
So the maximum height that the rocket will reach is 324 feet.
Therefore, the rocket will reach its maximum height of 324 feet after 4.5 seconds.
Step-by-step explanation:
hope its help<:
the angle has 4x and 6x what is the measurement of the angle
The sum of the measures of the two angles is 180 degrees, as they are angles of a straight line. The measure of the first angle is 4x = 72 degrees, and the measure of the second angle is 6x = 108 degrees.
What is angle ?
A measure of rotation of two crossing lines or planes in mathematics is called an angle. Angles are frequently defined as degrees or radians. Two lines or splines intersect at a location, known as the apex of the angle, to produce an angle. The edges of the angle are the two lines and line segments. According to its measurement, an angle can be categorized as: An acute angle is one that ranges from 0 to 90 degrees. Right arc: an angle that is 90 degrees in length. A measureable angle intermediate 90 and 180 degrees is referred to as an obtuse angle. 180 degree angle is referred to as a straight angle.
The sum of the measures of the two angles is 180 degrees, because they are angles of a straight line. Therefore:
4x + 6x = 180
Simplifying the left-hand side, we get:
10x = 180
Dividing both sides by 10, we obtain:
x = 18
So the measure of the first angle is:
4x = 4(18) = 72 degrees
And the measure of the second angle is:
6x = 6(18) = 108 degrees
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The complete question is: What is the measurement of an angle if its measure is 4x and another angle's measure is 6x?
find the value of x and y
Answer:
Try using the Symbolab calculator.
Step-by-step explanation:
Thats what I usually use and it works
At summer camp, 40 students are divided in two groups for swimming or hiking. Each camper flips a coin, where heads represents swimming and tails represents hiking.
Outcome Frequency
Swimming 12
Hiking 28
Compare the probabilities and determine which statement is true.
The theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 28 over 40.
The theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 12 over 40.
The theoretical probability of swimming, P(swimming), is 12 over 28, but the experimental probability is one half.
The theoretical probability of swimming, P(swimming), is 28 over 40, but the experimental probability is one half.
Answer:
Step-by-step explanation:
The theoretical probability of an event is the number of favorable outcomes over the total number of possible outcomes. In this case, the total number of possible outcomes is 2, since each camper can either swim or hike. The number of favorable outcomes for swimming is also 2 (heads on a coin flip). Therefore, the theoretical probability of swimming is 2/2 or 1/2.
The experimental probability of an event is the number of times the event occurred in the experiment divided by the total number of trials. In this case, there were 12 students who chose to swim out of 40 total students. Therefore, the experimental probability of swimming is 12/40 or 3/10.
Comparing the two probabilities, we see that the theoretical probability of swimming is 1/2 while the experimental probability of swimming is 3/10. Therefore, the statement "The theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 12 over 40" is true.
The theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 12 over 40.
The theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 12 over 40. (option b)
The theoretical probability of an event is calculated based on the assumption of equal likelihood for all possible outcomes. In this case, since each student flips a fair coin, there are two equally likely outcomes: heads (swimming) and tails (hiking). Therefore, the theoretical probability of swimming (P(swimming)) is 1/2, and the theoretical probability of hiking is also 1/2.
However, the experimental probability is determined by the actual outcomes observed in the experiment. According to the data provided, out of the 40 students, 12 students got heads (swimming) and 28 students got tails (hiking). To find the experimental probability of swimming, we divide the number of students who swam by the total number of students:
12/40 = 0.3.
The theoretical probability of swimming, P(swimming), is one half (0.5), but the experimental probability is 28 over 40 (0.7).
The theoretical probability of swimming, P(swimming), is one half (0.5), but the experimental probability is 12 over 40 (0.3).
The theoretical probability of swimming, P(swimming), is 12 over 28 (~0.4286), but the experimental probability is one half (0.5).
The theoretical probability of swimming, P(swimming), is 28 over 40 (0.7), but the experimental probability is one half (0.5).
Out of these options, the correct one is the theoretical probability of swimming, P(swimming), is one half, but the experimental probability is 12 over 40. (option b).
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Solve the radical equation √2-√36 - 2x = 6. Check for extraneous solutions.
The solution is |
The extraneous solution is
Enter the real solution first followed by the extarneous solution separated
by a comma.
For example, if real solution is 10 and the extraneous solution is 2.
Enter as 10,2.
Answer:
We have the equation:
√2 - √36 - 2x = 6
Simplifying the radicals we get:
√2 - 6 - 2x = 6
Adding 6 to both sides we get:
√2 - 2x = 12
Subtracting √2 from both sides we get:
-2x = 12 - √2
Dividing by -2 we get:
x = (6 - (1/√2))
Therefore, the real solution is:
x = (6 - (1/√2))
To check for extraneous solutions, we need to substitute this value of x back into the original equation and check if it satisfies the equation.
√2 - √36 - 2(6 - (1/√2)) = 6
Simplifying this we get:
-6 = 6
This is not true, so the solution x = (6 - (1/√2)) is extraneous.
Therefore, there is no real solution to the equation.
Find a1 in the geometric series Sn=3045, r=2/5, and an=120 Please help this is due at 11:59 pm TONIGHT! Remember to show work!
Answer:
a1 = 1875
Step-by-step explanation:
You want a1 for the geometric series that has r = 2/5, an = 120, and Sn = 3045.
Series relationsThe n-th term of the geometric series with first term a1 and common ratio r is ...
an = a1·r^(n-1)
The sum of the first n terms of the geometric series is ...
Sn = a1·(r^n -1)/(r -1)
ApplicationWe can use the expression for an to substitute for r^n in the sum equation:
[tex]a_n=a_1(r^n)(r^{-1})\\\\r^n=\dfrac{a_n}{a_1r^{-1}}=\dfrac{a_nr}{a_1}\\\\S_n=a_1\dfrac{r_n-1}{r-1}=a1\dfrac{\dfrac{a_nr}{a_1}-1}{r-1}\\\\S_n=\dfrac{a_nr-a_1}{r-1}\\\\a_1=a_nr-S_n(r-1)=a_nr+(1-r)S_n\\\\a_1=(120)\dfrac{2}{5}+(1-\dfrac{2}{5})(3045)=\dfrac{2\cdot120+3\cdot3045}{5}\\\\\boxed{a_1=1875}[/tex]
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Additional comment
The sum is of the first four terms:
1875 +750 +300 +120 = 3045
We can also find this by working backward. The previous term is 5/2 times the current term. We need to find the terms that have a sum of 3045.
120 +300 +750 +1875 +4687.5 +...
Clearly, 5 terms is too many. The sum of the first 4 terms of the backward series is 3045, so we know n=4 and the first term is 1875—the last term of our 4-term backward series.