The number you are looking for is 89.
Explanation:The number you are looking for is 89.
Here is how we can find it:
Since the number is between 80 and 100, we know it must have a tens digit of 8. The ones digit is two less than the tens digit, so it would be 6. Now we have the number 86, but we need to check if it is a prime number. A prime number is a number that is only divisible by 1 and itself. We can check if 86 is divisible by any number less than itself.Since we can divide 86 by 2 and get 43, it is not a prime number.Therefore, we have to keep increasing the ones digit by 1 until we find a prime number.The next number is 87, but it is divisible by both 3 and 29. Finally, we arrive at 89, which is only divisible by 1 and itself, making it a prime number.Learn more about Prime numbers here:https://brainly.com/question/35915729
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The graph of a proportional relationship passes through the points (4,3) and (x,9). Find x.
If x and y are proportional, then y : x = constant.
[tex]\dfrac{y}{x}=a\to y=ax[/tex]
We have (4, 3) and (x, 9). Substitute:
[tex]\dfrac{9}{x}=\dfrac{3}{4}[/tex] cross multiply
[tex]3x=(9)(4)[/tex]
[tex]3x=36[/tex] divide both sides by 3
[tex]\boxed{x=12}[/tex]
order 34 x 10^2,1.2x10^7,8.11×10-^3 and 435 from least to greatest
Answer:
The ascending order is [tex]8.11\times 10^{-3},435, 34\times 10^2, 1.2\times 10^7[/tex]
Step-by-step explanation:
First of all find the exact value of each number.
[tex]34\times 10^2=34\times 100=3400[/tex]
[tex]1.2\times 10^7=1.2\times 10000000=12000000[/tex]
[tex]8.11\times 10^{-3}=\frac{8.11}{1000}=0.00811[/tex]
The fourth number is 435.
Using the above calculation we can easily arrange these numbers form least to greatest.
The ascending order is
[tex]0.00811, 435, 3400, 12000000[/tex]
It can be written as
[tex]8.11\times 10^{-3},435, 34\times 10^2, 1.2\times 10^7[/tex].
Taryn is hosting a party at a restaurant the restaurant is charging her $140 to rent the space and $19 per guest if taryn wants to spend less then $615 which inequality could be used to solve for x the number of guests taryn can invite
Answer: 615= 140 + 19x
Step-by-step explanation:
19x represents the price per guess (x) how many people are coming. Good Luck!
To solve for the number of guests Taryn can invite, set up the inequality 140 + 19x < 615 and then solve for x.
Explanation:To solve for the number of guests Taryn can invite, we need to set up an inequality. Let x represent the number of guests. The total cost, C, can be calculated by adding the cost to rent the space, $140, to the cost per guest, $19, multiplied by the number of guests. So, the inequality can be written as 140 + 19x < 615.
To solve for x, we need to isolate it by subtracting 140 from both sides: 19x < 475. Finally, divide both sides by 19 to get the solution: x < 25.
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PLEASE HELP QUICK, GIVING BRAINLIEST
A cone-shaped hat is shown:
(below)
What is the approximate height of the hat?
3.51 centimeters, because height = Square root of the difference of 18 and 5.7
17.07 centimeters, because height = Square root of the difference of the squares of 18 and 5.7
18.88 centimeters, because height = Square root of the sum of the squares of 18 and 5.7
4.87 centimeters, because height = Square root of the sum of 18 and 5.7
This is classic Pythagorean solving:
Side^2 + Side^2 = Hypotenuse^2
The hypotenuse is the longest side of a triangle, and it connects the leg and the base usually. In this triangle, 18 is the hypotenuse -- so since we are solving for the unknown side, the equation is:
a^2 + 5.7^2 = 18^2 -- now solve
a^2 = 18^2 - 5.7^2
a^2 = 291.51
a = sqrt(291.51)
a = 17.07 -- because height is the square root of the difference of the squares of 18 and 5.7
The answer is choice B.
The correct answer is b. 17.07 centimeters, because height = Square root of the difference of the squares of 18 and 5.7.
To find the approximate height of the cone-shaped hat, we need to use the Pythagorean theorem. The height of the cone (h) can be found using the formula:
[tex]\[ h = \sqrt{r^2 - a^2} \][/tex]
where r is the slant height of the cone and a is the radius of the base of the cone.
Given that r = 18 cm and a = 5.7 cm, we substitute these values into the formula:
[tex]\[ h = \sqrt{18^2 - 5.7^2} \][/tex]
[tex]\[ h = \sqrt{324 - 32.49} \][/tex]
[tex]\[ h = \sqrt{291.51} \][/tex]
Now, we can approximate the square root of 291.51 to find the height:
[tex]\[ h \approx 17.07 \text{ cm} \[/tex]]
Therefore, the approximate height of the hat is 17.07 centimeters. The other options are incorrect because they do not correctly apply the Pythagorean theorem:
a. 3.51 centimeters: This is the square root of the difference of 18 and 5.7, which does not use the correct formula for the height of a cone.
c. 18.88 centimeters: This is the square root of the sum of the squares of 18 and 5.7, which would be the length of the slant height, not the height of the cone.
d. 4.87 centimeters: This is the square root of the sum of 18 and 5.7, which is not a correct application of the Pythagorean theorem for finding the height of a cone."
what is sin 45 degree?
Answer:
The answer is 1/square root 2
Step-by-step explanation:
Jaimie swims 12 the length of the swimming pool every 14 of a minute.
Enter the number of lengths of the swimming pool Jaimie swims per minute.
_[blank]_ lengths
Jaimie swims 12 the length of the swimming pool every 14 of a minute. Then in one minute, he will swim = 14/12=1.16 So, he will be able to swim 1.16 length in 1 minute.
What is the length in feet ?
Answer:
The length is 81 feet.
Step-by-step explanation:
If you plot the equations for both length and width on a graph, making them both equal to y of course, you find that they cross each other in two places. The places they cross are (81, 11) and (70, 0). It is impossible for the answer to have the width as 0 feet, so the answer is 81 feet.
Answer:
depends on the how many fee your talking
Step-by-step explanation:
What is the slope of the line?
4x−1=3y+5
Answer:
4/3 or D
Step-by-step explanation:
Let's get this equation into y=mx+b form.
m=slope b=y-int
Subtract 5 from both sides to isolate y on a side.
4x-6=3y
Divide both sides by 3 to isolate y on one side.
(4/3)x-2=y
y=(4/3)x-2
m=4/3
b=-2
So our slope is 4/3
solve for y.
enter your answer in the box.
y= __
Answer:
y = 10 PLEASE GIVE BRAINLIEST
Step-by-step explanation:
The angles of this triangle should all add up to 180 degrees.
There are 3 angles so 180 ÷ 3 = 60. Each angle will = 60 degrees.
To find y:
5y + 10 = 60
5y = 60 - 10
5y = 50
y = 50 ÷ 5
y = 10
olve for x. 3(3x - 1) + 2(3 - x) = 0 x = 3/7 x = -3/7 x = 1/3 x = -1/3
Answer:
x = -3/7
Step-by-step explanation:
We are given the following equation for which we have to solve for x:
[tex]3(3x - 1) + 2(3 - x) = 0[/tex]
So we will start by expanding the equation by multiplying the common factor with the brackets to get:
[tex]9x-3+6-2x=0[/tex]
Arranging the like terms together (variables on one side of the equation and constants on the other side) to get:
[tex]9x-2x=3-6\\\\7x=-3\\\\x=-\frac{3}{7}[/tex]
Therefore, x = -3/7.
Describe how the graph of y=x^2 would be shifted to produce a graph of y=2x^2+12x+3=0?
Answer:
The graph shifts 3 units left and 15 down. It narrows by a factor of 2 and is right side up.
Step-by-step explanation:
There are two ways of doing this. You can graph it, or you can put it in the vertex form by completing the square.
A quick way of putting this into the vertex form is write the general equation
y = a(x - h)^2 + k
h= - b/2a = - 12/4 = - 3
So far what you have is y = 2(x - - 3)^2 + ky = 2(x + 3)^2 + kk = f(x) = f(-3) = 2*(-3)^2 + 12(-3) + 3k = 2(9) - 36 + 3k = - 18 + 3 = - 15So the equation in vertex form is
y = 2(x + 3)^2 - 15
Description
y = 2(x + 3)^2 - 15
shifts 3 units to the left and 15 units down. It also narrows by a factor of 2. Finally, it is right side up.
Graph
Since you have seen the graph before, there is not much more to add. It just shows what has been written above.
Which of the following is the best definition of a circle?
Answer:
The set of all points on a plane equidistant to a given point
Step-by-step explanation:
"following"?
Answer:
The collection of all points in a plane that are the same distance from a given point.
Step-by-step explanation:
Woody wants to earn $250 to buy a bicycle by selling lemonade it cost him$0.05 dollars to make a glass of lemonade,which he sells for$0.25. how many drinks does he have to sell to earn$250
Answer:
1,250 drinks
Step-by-step explanation:
Woody spends $0.05 to make a glass of lemonade and makes $0.25 when he sells one, so his net gain is $0.20. 250 / .20 is 1,250.
Final answer:
Woody needs to sell 1,250 glasses of lemonade to earn the $250 required to buy a bicycle. Calculated by dividing the goal ($250) by the profit per glass ($0.20), which is the difference between the selling price ($0.25) and the cost price ($0.05).
Explanation:
Woody is operating a small business of selling lemonade and seeks to achieve a certain amount of total revenues to purchase a bicycle. To solve this, one must determine the profit per glass of lemonade sold, which is the selling price minus the cost to make it. Since Woody sells a glass for $0.25 and it costs $0.05 to make, the profit per glass is $0.25 - $0.05 = $0.20.
To calculate the total number of glasses Woody needs to sell to earn $250, divide his goal by the profit per glass. Thus:
Total number of glasses = ($250 / $0.20 per glass) = 1250 glasses
Hence, Woody must sell 1,250 glasses of lemonade to earn enough money to buy the bicycle.
Which of the following points lies in the solution
Answer:
B. (2, 0) - True, True, True
Step-by-step explanation:
A. (2, 3) - False
y ≤ 6x - 12 (False)
y ≤ 1/2x + 6 (False)
y ≥ -x - 4 (True)
B. (2, 0) - True, True, True
y ≤ 6x - 12 (True)
y ≤ 1/2x + 6 (True)
y ≥ -x - 4 (True)
C. (-3, 0) - False
y ≤ 6x - 12 (False)
y ≤ 1/2x + 6 (False)
y ≥ -x - 4 (True)
D. (1,1) - False
y ≤ 6x - 12 (False)
y ≤ 1/2x + 6 (False)
y ≥ -x - 4 (True)
Brainliest?
What is the product of −225 and −356 ? −915 −6730 613 915
Hello from MrBillDoesMath!
Answer:
80,100 (which is NONE of the provided answers)
Discussion:
-225 * -356 =
+ (225 * 356) = as the product of two negative numbers is positive
80,100
Regards,
MrB
P.S. I'll be on vacation from Friday, Dec 22 to Jan 2, 2019. Have a Great New Year!
The answer should be D. 9 1/5
what number is 45% of 180
Answer:
81
Step-by-step explanation:
45% × 180 = [tex]\frac{45}{100}[/tex] × 180 = 81
Answer:
The answer is 81
Step-by-step explanation:
You can do it following the next steps
1.- Place the the "45" by 100 = 0.45
2.- Multiply 0.45 times 180= 81
Anna picked 37 apples . She divided the apples equally among 7 baskets. How many apples are in each basket? How many are leftover?
Question
Anna picked 37 apples . She divided the apples equally among 7 baskets. How many apples are in each basket? How many are leftover?
Answer:
2Step-by-step explanation:
37 : 7 = 5 (2 left over)
check
(7 * 5) + 2 = 37
better solution in the picture
Answer:
5 apples in each basket with 2 left over
Step-by-step explanation:
37/7 =5 R 2
brainliest pls
Tools of Geometry:Question 2
Circle K has a center at (3,7) and has a radius of 1. Circle K' has a center of (7,–3) with a radius of 1. What is the transformation of circle K to circle K'?
Answer:
I think it's rotated 90 degrees clockwise
Step-by-step explanation:
The transformation of circle K to circle K' is a translation, specifically a movement of 4 units to the right and 10 units down, represented by T(4, -10).
Explanation:The transformation that maps circle K to circle K' is a translation. In a translation, every point of the object moves the same distance in the same direction. For circle K to circle K', this would be a shift 4 units to the right and 10 units down. The coordinates of the center of the circle move from (3,7) to (7,-3). We can calculate this movement by subtracting the initial coordinates from the final ones: (7-3, -3-7) = (4, -10). Therefore, the transformation would be written as T(4, -10).
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Irving can I remember the correct order of the five digits on his ID number he does remember that the ID number contains the digits 1.4,3,7,6 what is the probability that the first three digits of Irvings ID number will be odd numbers
Answer: [tex]\frac{27}{125}[/tex]
Step-by-step explanation:
Here the total numbers are 1, 4, 3, 7, 6
Since the total number of possible arrangement = [tex]5\times 5 \times5 \times 5 \times 5=5^5[/tex]
The total number of the odd numbers in the given numbers = 3
Thus the possible arrangement that the first three digits will be odd numbers = [tex]3\times 3\times 3\times 5\times 5=3^3\times 5^2[/tex]
Thus, the probability that the first three digits of Irvings ID number will be odd numbers = the possible arrangement that the first three digits will be odd numbers / total possible arrangement = [tex]\frac{3^3\times 5^2}{5^5} = \frac{3^2}{5^{5-2}}[/tex]
= [tex]\frac{3^3}{5^3} = \frac{27}{125}[/tex]
find the exact value of y
ΔABC and ΔACD are similar. Therefore the sides are in proportion:
[tex]\dfrac{y}{7}=\dfrac{7}{6}[/tex] multiply both sides by 7
[tex]y=\dfrac{49}{6}\\\\\boxed{y=\dfrac{1}{6}}[/tex]
Is 2 times the square root of 3 rational?
Answer:
Yes
Step-by-step explanation:
2 x √3 = 3.46410161514 , which is a rational number, because it is a real nomber
As an estimation we are told £3 is €4.
Convert £12.30 to euros.
Give your answer rounded to 2 DP.
Answer:
€16.40
Step-by-step explanation:
First, we need to divide £12.30 by 3 to find out how many groups of 3 there is:
12.3 ÷ 3 = 4.1
Now we can multiply this by 4 to find the euro equivalent:
4.1 * 4 = 16.4
So £12.30 is €16.40
Please help. I really need help with this so please help.
Trisha needs to make at least 50 gift bags for an event. Each gift bag will contain at least 1 thumb drive or 1 key chain. She wants to use at least 5 times as many key chains as thumb drives. She has 25 thumb drives and 200 key chains.
Let x represent the number of key chains. Let y represent the number of thumb drives.
Write an inequality for this situation.
To ensure Trisha has at least 5 times as many key chains as thumb drives in each gift bag, the inequality x ≥ 5y can be used, where x is the number of key chains and y is the number of thumb drives. Considering she needs to make at least 50 bags, the inequality x + y ≥ 50 is also relevant.
For Trisha to satisfy the condition of having at least 5 times as many key chains as thumb drives in each gift bag, the following inequality can represent her situation:
x ≥ 5y
Here, x is the number of key chains and y is the number of thumb drives. Trisha must also create at least 50 gift bags, so another inequality to consider is the total number of gift bags:
x + y ≥ 50
Since Trisha has 25 thumb drives (y) and 200 key chains (x), she is limited by the number of thumb drives. The maximum number of gift bags she can create with one thumb drive in each is 25 bags, which means she can use up to 125 key chains (5 times 25). Thus, Trisha can satisfy the requirement by using all 25 thumb drives and up to 125 key chains, ensuring there's a minimum of one thumb drive or key chain per bag.
Find the distance of line AB if A(3, 5) and B(-3,-3).
8
6
12
10
The distance of line AB if A(3, 5) and B(-3,-3) is D.10.
[tex]\sqrt[/tex] (5--3)^2 + (3-(-3))^2 = V100 = 10.
How to calculate the AB distance?
To calculate the distance AB between points A (x1, y1) and B (x2, y2), first draw a right triangle with segment ¯AB in the hypothalamus. AC is a horizontal distance, so it's just a difference in x-coordinates: | (x2-x1) |. Similarly, BC is the vertical distance | (y2-y1) |. ..
The vertical distance of the line (Ax + By + c = 0) from point
is equal to | Ax'+ By' + c√A2 + B2 |. Where (x', y') are the coordinates of the point. Since we need the distance of the line from the origin, the coordinates will be (0,0), which is the vertical distance of the line from the origin.
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What is the length of Line BC?
The length of Line BC is 10
What is a triangle?A figure bounded by 3 sides and all the internal angles add up to 180 ° is called a triangle.
What are similar triangles?Triangles having the same corresponding angles measures and proportional side lengths are called similar triangles.
How to find the length of the line BC?Considering Δ AXY and ΔABC
∠AXY = ∠ACB (corresponding angles are equal)
∠AYX = ∠ABC (corresponding angles are equal)
∠A is common in both the triangles
∴ We can say the triangles are similar .
Now, let H be the height of ΔABC and h be the height of ΔAYX
So, we can say, [tex]\frac{XY}{BC} = \frac{h}{H}[/tex]
Now, we know that area of the triangle can be found with the help of the formula (1/2)x(base)x(height)So, [tex]\frac{AYX}{ABC} = \frac{7.5}{30}[/tex]
⇒ [tex]{\frac{\frac{1}{2} (XY) h }{\frac{1}{2}(BC) H } = \frac{1}{4}[/tex]
Now, substituting XY = 5 and [tex]\frac{XY}{BC} = \frac{h}{H}[/tex] in the above equation, we get
[tex]\frac{25}{BC^{2} } = \frac{1}{4}[/tex]
⇒ BC² = 100
⇒ BC = ± 10
Since BC is the length , so it cannot be negative.
So, the negative value is rejected.
∴ BC = 10
Option B is correct.
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Which ordered pair is the solution to the system of equations? PLS HELP
−3x+4y=−20y=x−4
(4, 8)
(0, −5)
(−2, −6)
(−4, −8)
Answer:
The correct answer is (−4, −8)
Step-by-step explanation:
You have to substitute!!! :)
−3x + 4y = −20 y = x − 4
-3(4) + 4(8) = -20 8 = 4 - 4
-12 + 32 = -20 8 = 0
20 = 20
−3x + 4y = −20 y = x − 4
-3(0) + 4(-5) = -20 -5 = 0 - 4
0 + 20 = -20 -5 = -4
20 = -20
−3x + 4y = −20 y = x − 4
-3(-2) + 4(-6) = -20 -6 = -2 - 4
6 - 24 = -20 -6 = -6
-18 = -20
−3x + 4y = −20 y = x − 4
-3(-4) + 4(-8) = -20 -8 = -4 - 4
12 - 32 = -20 -8 = -8
-20 = -20
Answer:
(-4,-8)
Step-by-step explanation:
This a system of equation that have two variables x and y and we have two equations that we have to solve simultaneously
−3x+4y=−20 ....(1)
y=x−4 ....(2)
We can substitute equation (2) and (1)
-3x+4y = −20
-3x+4(x−4) = −20
-3x + 4x -16 = -20
x = -4
Now we need to find the value of y
y=x−4 = -4 - 4 = -8
(-4,-8)
Alexis counted 16 commercials while watching 30 minutes of TV. At this rate, how many commercials would play in a day?
30 minutes is half an hour.
so in 30 minutes is 16 commercials, so in 1 hour is twice as many, 32 commercials.
there are 24 hours in a day, so 32 commercials 24 times, 32 * 24.
a spinner divided into four equal parts, a, b, c and d is spun then is followed by a roll of a standard six sided die.
how many total outcomes are possible?
how many outcomes have an ‘A’ and an odd number ?
Answer:
Possible number of outcomes = 24
Number of outcomes have an 'a' and an odd number = 3
Step-by-step explanation:
The total outcomes are as follows:
S = {(a, 1), (a, 2), (a, 3), (a, 4), (a, 5), (a, 6),
(b, 1), (b, 2), (b, 3), (b, 4), (b, 5), (b, 6),
(c, 1), (c, 2), (c, 3), (c, 4), (c, 5), (c, 6),
(d, 1), (d, 2), (d, 3), (d, 4), (d, 5), (d, 6)}
n(S) = 24
Hence, possible number of outcomes = 24
The outcomes have an 'a' and an odd number are:
E = {(a, 1), (a, 3), (a, 5)}
n(E) = 3
Hence, there are 3 outcomes have an 'a' and an odd number.
p=10-2.5c, find p when c = 3.2 and then find c when p = 85
Final answer:
To find the value of p when c = 3.2, substitute c = 3.2 into the expression for p. The value of p is 2. To find the value of c when p = 85, substitute p = 85 into the expression for p and solve for c. The value of c is -30.
Explanation:
To find the value of p when c = 3.2, we substitute c = 3.2 into the expression for p.
Substituting c = 3.2 into p = 10 - 2.5c, we get p = 10 - 2.5 * 3.2 = 10 - 8 = 2.
So, p = 2 when c = 3.2.
To find the value of c when p = 85, we substitute p = 85 into the expression for p.
Substituting p = 85 into p = 10 - 2.5c, we get 85 = 10 - 2.5c.
Then, we solve for c: 2.5c = 10 - 85 = -75.
Dividing both sides by 2.5, we get c = -75/2.5 = -30.
So, c = -30 when p = 85.
Express the number 220 as the sum of four numbers that form a geometric progression such that the third term is greater than the first by 44.
Answer:
Step-by-step explanation:
The first four terms of geometric series is:
[tex]a,ar,ar^2,ar^3[/tex]
Since, we have given information that the third number is greater than 44 that means:
[tex]ar^2=a+44[/tex]
Above equation can be rewritten as:
[tex]a(r^2-1)=44[/tex]
Now, using:
[tex]a^2-b^2=(a+b)(a-b)[/tex]
Here, a=r,b=1
[tex]a(r+1)(r-1)=44[/tex] (1)
The sum of first four terms is:
[tex]a+ar+ar^2+ar^3=220[/tex]
[tex]a(1+r+r^2+r^3)=220[/tex]
[tex]a(1(1+r)+r^2(1+r))=220[/tex]
[tex]a((1+r)(1+r^2))=220[/tex] (2)
Divide equation (2) by (1) we get:
[tex]\frac{r^2+1}{r-1}=\frac{220}{44}[/tex]
[tex]r^2+1=5r-5[/tex]
[tex]\Rightarrow r^2-5r+6=0[/tex]
[tex]\Rightarrow r^2-3r-2r+6=0[/tex]
[tex]\Rightarrow r(r-3)-2(r-3)=0[/tex]
[tex]\Rightarrow (r-2)(r-3)=0[/tex]
[tex]\Rightarrow r=2,3[/tex]
CASE1: When r=2 in [tex]ar^2=a+44[/tex]
[tex]a(4)=a+44[/tex]
[tex]3a=44[/tex]
[tex]a=\frac{44}{3}[/tex]
CASE2:When r=3 in [tex]ar^2=a+44[/tex]
[tex]a(3)^2=a+44[/tex]
[tex]\Rightarrow 9a=a+44[/tex]
[tex]\Rightarrow 8a=44[/tex]
[tex]\Rightarrow a=\frac{44}{8}=\frac{11}{2}[/tex]
The series becomes:
From CASE1: [tex]\frac{44}{3},\frac{44\cdot 2}{3},\frac{44\cdot 2^2}{3},\frac{44\cdot 2^3}{3}[/tex]
[tex]\Rightarrow \frac{44}{3},\frac{88}{3},\frac{176}{3},\frac{352}{3}[/tex]
From CASE2: [tex]\frac{11}{2},\frac{11\cdot 3}{2},\frac{11\cdot 3^2}{2},\frac{11\cdot 3^3}{2}[/tex]
[tex]\Rightarrow \frac{11}{2},\frac{33}{2},\frac{99}{2},\frac{297}{2}[/tex]
A geometric progression is characterized by a common ratio.
The terms of the progression are: [tex]\mathbf{T_n =5.5, 16.5, 49.5,148.5}[/tex] or [tex]\mathbf{T_n =\frac{44}{3}, \frac{88}{3}, \frac{176}{3},\frac{352}{3}}[/tex]
The given parameters are:
[tex]\mathbf{S_4 = 220}[/tex]
[tex]\mathbf{T_3 = T_1 + 44}[/tex]
The third term is represented as:
[tex]\mathbf{T_3 = ar^2}[/tex]
The first term is:
[tex]\mathbf{T_1 = a}[/tex]
So, we have:
[tex]\mathbf{ar^2 = a + 44}[/tex]
Subtract a from both sides
[tex]\mathbf{ar^2 - a = 44}[/tex]
Factor out a
[tex]\mathbf{a(r^2 - 1) = 44}[/tex]
Express as difference of two squares
[tex]\mathbf{a(r + 1)(r - 1) = 44}[/tex]
Make r + 1 the subject
[tex]\mathbf{r +1 = \frac{44}{a(r -1)}}[/tex]
Recall that:
[tex]\mathbf{S_4 = 220}[/tex]
This gives:
[tex]\mathbf{a + ar + ar^2 + ar^3 = 220}[/tex]
Factor out a
[tex]\mathbf{a(1 + r + r^2 + r^3) = 220}[/tex]
Factor out r^2
[tex]\mathbf{a(1 + r + r^2(1 + r)) = 220}[/tex]
Rewrite as:
[tex]\mathbf{a(1(1 + r) + r^2(1 + r)) = 220}[/tex]
Factor out 1 + r
[tex]\mathbf{a((1 + r^2)(1 + r)) = 220}[/tex]
Substitute [tex]\mathbf{r +1 = \frac{44}{a(r -1)}}[/tex] in [tex]\mathbf{a((1 + r^2)(1 + r)) = 220}[/tex]
[tex]\mathbf{a((1 + r^2)\times \frac{44}{a(r -1)} = 220}[/tex]
[tex]\mathbf{((1 + r^2)\times \frac{44}{(r -1)} = 220}[/tex]
Divide both sides by 44
[tex]\mathbf{(1 + r^2)\times \frac{1}{(r -1)} = 5}[/tex]
Multiply through by r - 1
[tex]\mathbf{1 + r^2 = 5r - 5}[/tex]
Collect like terms
[tex]\mathbf{r^2 - 5r + 5 +1 = 0}[/tex]
[tex]\mathbf{r^2 - 5r + 6 = 0}[/tex]
Expand
[tex]\mathbf{r^2 - 2r - 3r + 6 = 0}[/tex]
Factorize
[tex]\mathbf{r(r - 2) - 3(r - 2) = 0}[/tex]
Factor out r -2
[tex]\mathbf{(r - 3)(r - 2) = 0}[/tex]
So, we have:
[tex]\mathbf{r = 3\ or\ r = 2}[/tex]
Calculate a using: [tex]\mathbf{a(r^2 - 1) = 44}[/tex]
When r = 3
[tex]\mathbf{a(3^2 -1) = 44}[/tex]
[tex]\mathbf{a(9 -1) = 44}[/tex]
[tex]\mathbf{8a = 44}[/tex]
[tex]\mathbf{a = 5.5}[/tex]
When r = 2
[tex]\mathbf{a(2^2 -1) = 44}[/tex]
[tex]\mathbf{a(4 -1) = 44}[/tex]
[tex]\mathbf{3a = 44}[/tex]
[tex]\mathbf{a = \frac{44}{3}}[/tex]
The nth term of a GP is:
[tex]\mathbf{T_ = ar^{n-1}}[/tex]
So, the terms of the progression are:
[tex]\mathbf{T_n =5.5, 16.5, 49.5,148.5}[/tex]
or
[tex]\mathbf{T_n =\frac{44}{3}, \frac{88}{3}, \frac{176}{3},\frac{352}{3}}[/tex]
Read more about geometric progressions at:
https://brainly.com/question/18109692