Answer:
Hence,
[tex]S_5=1162.5[/tex]
Step-by-step explanation:
We are asked to evaluate:
[tex]S_5[/tex]
We are given a geometric series.
( since each term of the series are in geometric progression as each term is half of the previous term of the series.
i.e. we have a common ratio of 1/2 )
Also, we know that sum of n terms in geometric progression is given by:
[tex]S_n=a(\dfrac{1-r^n}{1-r})[/tex]
where r is the common ratio.
a is the first term of the series.
Here we have:
[tex]a=600\ ,\ r=\dfrac{1}{2}[/tex]
Hence,
[tex]S_5=600(\dfrac{1-(\dfrac{1}{2})^5}{1-\dfrac{1}{2}})\\\\\\S_5=600(\dfrac{2^5-1}{2^4}})\\\\\\S_5=600(\dfrac{31}{16})\\\\\\S_5=1162.5[/tex]
The sum of the first five terms of the geometric progression 600 + 300 + 150 + … is 1162.5.
What is the ratio of geometric progression?A geometric progression is the series of numbers where the ratio of the two consecutive numbers is always the same.
As it is given to us the sequence 600, 300, 150 is a geometric progression. Therefore, the first term of the progression is 600, while the ratio between the two terms are,
[tex]r = \dfrac{300}{600} = \dfrac{1}{2}[/tex]
Now, we know that the sum of the geometric sequence of the first 5 terms can be written as where the value of the n is 5,
[tex]S=a\dfrac{(1-r^n)}{(1-r)}[/tex]
[tex]S=600\dfrac{(1-(\dfrac{1}{2})^n)}{(1-(\dfrac{1}{2}))}[/tex]
[tex]S = 1162.5[/tex]
Hence, the sum of the first five terms of the geometric progression 600 + 300 + 150 + … is 1162.5.
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Which of the following is not a way to represent the solution of the inequality 2(x − 1) less than or equal to 10? (1 point)
A. x less than or equal to 6
B. 6 greater than or equal to x
C. A number line with a closed circle on 6 and shading to the left
D. A number line with a closed circle on 6 and shading to the right
A and B are the same and C is proven wrong due to the equality sign lkeaving only D to be correct
what is the reciprocal of 8-3i
The reciprocal of the complex number 8-3i is found by dividing its complex conjugate (8+3i) by the modulus squared of 8-3i, which is 73. The result is 8/73 + 3i/73.
Explanation:The reciprocal of a complex number is the complex conjugate of that number divided by the modulus squared of the original number. In the case of the complex number 8-3i, its complex conjugate is 8+3i. To find the reciprocal of 8-3i, you would divide this complex conjugate by the modulus squared of 8-3i.
First, calculate the modulus squared of 8-3i:
Modulus of 8-3i = sqrt(82 + (-3)2) = sqrt(64 + 9) = sqrt(73).Modulus squared = (sqrt(73))2 = 73.Then, divide the complex conjugate by this modulus squared to get the reciprocal:
Reciprocal of 8-3i = (8+3i) / 73 = 8/73 + 3i/73.
f-7 over g equals h, sole for f
the polynomial (x - 2) is a factor of the polynomial 3x2 - 8x + 2.
First, let's factor 3x²-8x+2
Looking at wee can see that the first term is and the last term is where the coefficients are 3 and 2 respectively.
Now multiply the first coefficient 3 and the last coefficient 2 to get 6. Now what two numbers multiply to 6 and add to the middle coefficient -8? Let's list all of the factors of 6:
Factors of 6:
1,2,3,6
-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 6
1*6
2*3
(-1)*(-6)
(-2)*(-3)
note: remember two negative numbers multiplied together make a positive number
Now, which of these pairs add to -8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -8
None of these pairs of factors add to -8.
So the expression 3x²-8x+2 cannot be factored.
So (x - 2) is NOT a factor of 3x²-8x+2
So the statement is false
Answer:
B.false
Step-by-step explanation:
apexs
Openstudy using synthetic division, what is the quotient (2x3 − 2x − 12) ÷ (x − 2)?
If the probability of an event is 0.7 repeating, what are the odds against the event
930 miles to travel with a car that gets 22 miles per gallon. How much will it cost to travel if gas costs 2.03 a gallon?
which of the following are not necessary when proving that the diagonals of a rectangle are congruent? check all that apply
Answer:
The correct options are B and D.
Step-by-step explanation:
A quadrilateral is called rectangle if the opposite sides are congruent and parallel to each other. All interior angles are right angle and congruent.
To prove that the diagonals of a rectangle are congruent the necessary conditions are
1. Opposite sides of a rectangle are congruent.
2. All right angles are congruent.
Therefore options A and C are necessary conditions. Option A and C are incorrect.
The opposite sides of a rectangle are parallel to each other and two parallel lines never intersect each other.
Therefore the opposite sides are not perpendicular to each other. Option D is correct.
The angle whose measure is more than 90 degree is called an obtuse angle.
Since all interior angles of a rectangle are right angle and congruent, therefore there is no obtuse angle. So, condition B is unnecessary. Option B is correct.
PLEASE HELP!! 25 POINTS
A rectangle has an area of 102 cm2. The length of the rectangle is 17 cm.
What is the perimeter of the rectangle?
Which of the following are arithmetic sequences? Check all that apply.
A. 1, 1, 2, 5, 8, 13
B. 5, 5, 5, 5, 5
C. 3, 6, 9, 12, 15
D. 2, 4, 8, 16, 32
Answer:
The correct option is B) 5, 5, 5, 5, 5 and C) 3, 6, 9, 12, 15 are arithmetic sequences
Step-by-step explanation:
Arithmetic sequence is: a, a+r, a+2r, a+3r, a+4r, ....... where r is common difference
Check part A) 1, 1, 2, 5, 8, 13
In which [tex]a_1=1,\, a_2=1,\,a_3=2,\,a_4=5[/tex]
common difference [tex]d=a_n+1 -a_n[/tex]
[tex]d=a_2 -a_1 =1-1=0 [/tex]
[tex]d=a_3 -a_2 =2-1=1 [/tex]
since common difference is not same
so, sequence 1, 1, 2, 5, 8, 13 is not arithmetic sequences.
Check part B) 5, 5, 5, 5, 5
In which [tex]a_1=5,\, a_2=5,\,a_3=5,\,a_4=5[/tex]
common difference [tex]d=a_n+1 -a_n[/tex]
[tex]d=a_2 -a_1 =5-5=0 [/tex]
[tex]d=a_3 -a_2 =5-5=0 [/tex]
since common difference is same
so, sequence 5, 5, 5, 5, 5 is arithmetic sequences.
Check part C) 3, 6, 9, 12, 15
In which [tex]a_1=3,\, a_2=6,\,a_3=9,\,a_4=12[/tex]
common difference [tex]d=a_n+1 -a_n[/tex]
[tex]d=a_2 -a_1 =6-3=3 [/tex]
[tex]d=a_3 -a_2 =9-6=3 [/tex]
since common difference is same
so, sequence 3, 6, 9, 12, 15 is arithmetic sequences.
Check part D) 2, 4, 8, 16, 32
In which [tex]a_1=2,\, a_2=4,\,a_3=8,\,a_4=16[/tex]
common difference [tex]d=a_n+1 -a_n[/tex]
[tex]d=a_2 -a_1 =4-2=2 [/tex]
[tex]d=a_3 -a_2 =8-4=4 [/tex]
since common difference is not same
so, sequence 2, 4, 8, 16, 32 is not arithmetic sequences.
Therefore, the correct option is B) 5, 5, 5, 5, 5 and C) 3, 6, 9, 12, 15 are arithmetic sequences
Which is an equation of a circle with center (2, -1) that passes through the point (3, 4)?
1. (x + 2)2 + (y - 1)2 = 26
2. (x + 2)2 + (y - 1)2 = 13
3. (x - 2)2 + (y + 1)2 = 26
4. (x - 2)2 + (y + 1)2 = 13
The number of points awarded in each round of a game can be represented by f(x) = 2x, where x represents the number of the round.
What is x when f(x) = 16?
Answer:
x=4; in round 4, 16 points will be rewarded
How to algebraically find the limit of a function as x approaches infinity?
Well... One way you can do this is by testing a set of arrays and see the trend. If I chose to find what y1 is in (100, y1) and what y2 is in (101, y2), I would find the difference between y2 and y1. If y2 - y1 is positive, this means there is a positive relationship and y is also approaching POSITIVE infinity. A negative relation means that it is approaching NEGATIVE infinity. However, it could be approaching a single number like "4" for instance, and you just need to plug in the right number of data sets to make that educated guess.
Formula Example:
5 + 1 / (x + 1) will always approach 5 because "1 / (x + 1) will approach 0".
Hope this helps.
To algebraically find limit of a function as "x" approaches infinity, identify highest degree term, divide by "x", and evaluate the resulting expression.
To algebraically find the limit of a function as x approaches infinity, we can follow the below steps :
(i) Identify the highest degree term: Determine the term with the highest power of x in the function.
(ii) Ignore lower degree terms: If the highest degree term is dominant, ignore lower degree terms and constants.
(iii) Divide by x: Divide every term in the function by x raised to the power of the highest degree.
(iv) Evaluate the limit: Examine the resulting expression as x approaches infinity. If the expression tends to a finite value, that is the limit. If the expression diverges (goes to infinity or negative infinity), then the limit does not exist.
By simplifying the function and observing the behavior of the resulting expression, we can algebraically determine the limit of the function as x approaches infinity.
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If the letters a, b, c, d, e, and f are to be used in a five-letter code, how many different codes are possible if repetitions are not permitted?
There are 150 marigold plants in a back yard. Each month, the number of marigold plants decreases by 15%. There are 125 sunflower plants in the back yard. Each month, 8 sunflower plants are removed. Part A: Write functions to represent the number of marigold plants and the number of sunflower plants in the back yard throughout the months. (4 points) Part B: How many marigold plants are in the back yard after 3 months? How many sunflower plants are in the back yard after the same number of months? (2 points) Part C: After approximately how many months is the number of marigold plants and the number of sunflower plants the same? Justify your answer mathematically. (4 points)
To solve this problem, let us first assign variables. Let us say that:
X = number of marigold plants
Y = number of sunflower plants
n = number of months
We can see that in the given problem, X is decreasing by a percentage, this means that we have to set-up a geometric equation while for Y the decrease is linear so we set-up an arithmetic equation.
Part A.
For marigold plants X, a geometric sequence has a general form of:
X = Xo * (1 + r)^n
where r = -15% = -0.15 (negative since it is decreasing)
Xo = the initial amount of marigold plants = 150
X = 150 * (1 – 0.15)^n
X = 150 (0.85)^n
For the sunflower plants Y, an arithmetic sequence has a general form of:
Y = Yo + d * n
where d = -8 and Yo = 125
Y = 125 – 8 n
Part B. For n = 3
X = 150 (0.85)^3 = 92.12 = 92
Y = 125 – 8 (3) = 101
Part C. From Part B we see that the two values are very far from each other when n = 3, therefore they must be similar when n < 3. So we try n = 2
X = 150 (0.85)^2 = 108.38 = 108
Y = 125 – 8 (2) = 109
Therefore the two plants have approximately similar amount after 2 months.
The function below shows the number of car owners f(t), in thousands, in a city in different years t: f(t) = 0.25t2 − 0.5t + 3.5 The average rate of change of f(t) from t = 2 to t = 6 is ______ thousand owners per year. Answer for Blank 1:
What is the equation of the blue graph?
A captain of a fishing boat pays each member of his crew $20, plus $3 per fish caught. How much would a crew member earn for catching 40 fish?
Answer: C) $140
20 times 3 plus 20
A crew member would earn $140 for catching 40 fish, calculated by adding the base pay of $20 to the additional pay of $3 per fish caught for the total number of fish.
To calculate how much a crew member would earn for catching 40 fish, given that the captain of the fishing boat pays each crew member a base amount of $20 plus an additional $3 per fish caught. To find the total earnings, we use the formula:
Total earnings = Base pay + (Pay per fish × Number of fish caught)
Plugging the given numbers into the formula, we get:
Total earnings = $20 + ($3 × 40)
Total earnings = $20 + $120
Total earnings = $140
Therefore, a crew member would earn $140 for catching 40 fish.
The trees at a national park have been increasing in numbers. There were 1,000 trees in the first year that the park started tracking them. Since then, there has been one fifth as many new trees each year. Create the sigma notation showing the infinite growth of the trees and find the sum, if possible.
1 1000
2 200
3 40
Answer:
A
Step-by-step explanation:
Find the value of X and Y. Show your work.
find multiplicative inverse of 2+3i / 3-2i
A square with an area of 64 in2 is rotated to form a cylinder. What is the radius of the cylinder?
A bag contains 21 red marbles, 22 green marbles, 24 orange marbles, and 10 yellow marbles. You choose a marble randomly from the bag.
What is the approximate probability that you will choose a green or yellow marble?
A. 0.358
B. 0.435
C. 0.416
D. 0.538
21 +22+24+10 = 77 total marbles
22 +10 =32 green and yellow marbles
32/77 probability of choosing a green or yellow one
32/77 = 0.416
answer is C. 0.416
If x = 3 is a zero of the polynomial function f(x) = 2x3 + x2 − 25x + 12, which of the following is another zero of f(x)? ASAP
The other zeros of the polynomial if 3 is a zero is -4 and 1/2
If x = 3 is a zero of the polynomial function f(x) = 2x3 + x2 − 25x + 12, then x - 3 is a factor.
To get the other factor we will take the division of the polynomial and the factor to have:
2x^3 + x^2 − 25x + 12/x - 3 = 2x² + 7x - 4
Factorize 2x² + 7x - 4
2x² + 7x - 4 = 0
2x² + 8x - x- 4 = 0
2x(x+4)-1(x+4) = 0
(x+4)(2x-1) = 0
x = -4 and 1/2
Hence the other zeros of the polynomial if 3 is a zero is -4 and 1/2
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The diameter of a beach ball is 10 inches. How many cubic inches of air can the beach ball hold? Use 3.14 for pie . Round to the nearest tenth of a cubic inch
volume = 4/3 x PI x r^3
r = 10/2 =5
V = 4/3 x 3.14 x 5^3 = 523.333
523.3 cubic inches
What is the measure of ABC?
Answer: Measure of <ABC=65°.
Step-by-step explanation:
The intercepted arc Theorem says :Angle made on circumference if half the measure of the intercepted arc.
In the given circle measure of arc is 130° The angle intercepted by arc AC is <ABC.
m<ABCwill be equal to Half of measure of arc AC
M<ABC=[tex]\frac{1}{2} of 130=65[/tex]
the area of a rectangle with width x and length 7x is 7x^2 what does the coefficient 7 mean in terms of the problem
Find the volume of each figure to the nearest tenth. Show your work.
A key code must contain 6 numerals. There are 10 numerals available. Using these numerals, how many different key codes may be created? A. 151,200 B. 340 C. 210 D. 3,480
Answer: Option A, 151,200 possible combinations.
Explanation:
The code must contain 6 numbers, and there are a total of 10 numbers avaible. Here we must assume that the numbers must be different, because of the word "avaible" and the options avaible.
This means that for the first number, we have 10 options, for the second number, we have 9 options (because we already took 1), in the third number, we have 8 options, and so on.
The total number of combinations is equal to the product between the number of options for each number in the code.
10*9*8*7*6*5 = 151,200
Then the right option is A.
Hillary rolls 2 number cubes numbered 1 through 6 while playing her favorite board game. She will get a second turn if she rolls a sum that is an odd number greater than 10. What are Hillary's chances of getting a second turn when she rolls the number cubes?