The sum of the first 10 terms of an arithmetic series is 100 and the sum of next 10 300 .Find the series.
Answers
Let a be the first term in the sequence, and d the common difference between consecutive terms. If aₙ denotes the n-th term in the sequence, then
a₁ = a
a₂ = a₁ + d = a + d
a₃ = a₂ + d = a + 2d
a₄ = a₃ + d = a + 3d
and so on, up to the n-th term
aₙ = a + (n - 1) d
The sum of the first 10 terms is 100, and so
\(\displaystyle \sum_{n=1}^{10} a_n = 100 \\ \sum_{n=1}^{10} (a + (n-1)d) = 100 \\ (a-d) \sum_{n=1}^{10} 1 + d \sum_{n=1}^{10} n = 100 \\ 10a+45d = 100\)
where we use the well-known sum formulas,
\(\displaystyle \sum_{n=1}^N 1 = 1 + 1 + 1 + \cdots + 1 = N\)
\(\displaystyle \sum_{n=1}^N n = 1 + 2 + 3 + \cdots + N = \frac{N(N+1)}2\)
The sum of the next 10 terms is 300, so
\(\displaystyle \sum_{n=11}^{20} a_n = 300 \\ (a-d) \sum_{n=11}^{20} 1 + d \sum_{n=11}^{20} n = 300 \\ (a-d) \left(\sum_{n=1}^{20} 1 - \sum_{n=1}^{10} 1\right) 1 + d \left(\sum_{n=1}^{20} n - \sum_{n=1}^{10} n\right) = 300 \\ 10a+145d = 300\)
Solve for a and d. Eliminating a gives
(10a + 145d) - (10a + 45d) = 300 - 100
100d = 200
d = 2
and solving for a gives
10a + 145×2 = 300
10a = 10
a = 1
So, the given sequence is simply the sequence of positive odd integers,
{1, 3, 5, 7, 9, …}
given recursively by the relation
\(\begin{cases}a_1 = 1 \\ a_n = a_{n-1} + 2 & \text{for }n>1\end{cases}\)
and explicitly by
\(a_n = 1 + 2(n-1) = 2n - 1\)
for n ≥ 1.
Related Questions
70 students choose to attend one of three after school activities: football, tennis or running. There are 30 boys. 13 students choose football, of which 9 are girls. 31 students choose tennis. 9 girls choose running. A student is selected at random. What is the probability this student chose running? Give your answer in its simplest form.
Answers
Answer:
9/70
Step-by-step explanation:
Given that:
70 students choose to attend one of three after school activities: football, tennis or running.
There are 30 boys. 13 students choose football, of which 9 are girls. 31 students choose tennis. 9 girls choose running.
A student is selected at random.
What is the probability this student chose running?
Since it asking for the probability of this student (randomly select) chose running. And we can see that there are 9 girls choose running.
Therefore, we also know 70 students is the total
Hence, the answer is 9/70
9/70 is in simplest form already.
~lenvy~
Answer:
9/70
Step-by-step explanation:
Important info:
70 students choose to attend one of three after school activities: football, tennis or running.There are 30 boys. 13 students choose football, of which 9 are girls. 31 students choose tennis. 9 girls choose running.A student is selected at random.What is the probability this student chose running?Solution:
Probability of this student (randomly select) chose running. And we can see that there are 9 girls choose running.
Thus, 70 students is the total
Therefore, the answer is 9/70
9/70 can't be simplify.
[RevyBreeze]
this is what i need help with so please help
Answers
Answer:
The temperature farthest from zero is 5
Step-by-step explanation:
the reason for this is because it doesn't matter whether that temperature is negative or positive because all the question is asking is...."how far from zero"....and would -4 be farther from zero....or would 5 be farther from zero. How I know this is absolute value is always positive. And absolute value of -4....is 4.....and absolute value of 5....is 5.
And 5 is 1 temperature more away from zero than -4.
Answer:
5 is the farthest from 0
-4 is the coldest temperature
5 is the warmest temperature
Step-by-step explanation:
The farthest from 0 could be negative or positive. It's just whichever one has the greatest absalute value (if you know that term, if not...), whichever one you would put up more fingers to count to.
In this case, according to the number line, the farthest from 0 into the nagatives is -4, but the farthest from 0 in the positives is 5. This means that 5 is the farthest from 5 because 5 takes 5 of your fingers to count to, whule -4 (yes, even though it's negative) takes just 4 fingers. This is a good way to visualize it until you can wrap your head around it!
The coldest is the lowest number. In this case, it's below zero. Below zero means negatives. In this case, -4 is the lowest number.
The warmest is the highest number. In this case, it's above zero. That means it's a positive number. 5 is the highest number here.
Suppose f(x) is a function such that if p
O f(x) can be odd or even.
O f(x) can be odd but cannot be even.
O f(x) can be even but cannot be odd.
O f(x) cannot be odd or even.
Answers
A two-child family is selected at random. Let B denote the event that at least one child is a boy, and let D denote the event that
the genders of the two children differ. Find the union of B and D. (1 point)
The union is (bb, gg); the first letter denotes the gender of the firstborn child, and the second letter denotes the gender of the
second child.
The union is (bg. gb); the first letter denotes the gender of the firstborn child, and the second letter denotes the gender of the
second child.
The union is {bb, bg, gb, gg); the first letter denotes the gender of the firstborn child, and the second letter denotes the gender of
the second child.
The union is (bb, bg. gb); the first letter denotes the gender of the firstborn child, and the second letter denotes the gender of the
second child.
Answers
With the help of probability we can say that, DUM={GG,BB,GB,BG).
What is Probability?Probability is the concept that describes the likelihood of an event occurring.
In real life, we frequently have to make predictions about how things will turn out.
We may be aware of the result of an occurrence or not.
When this occurs, we state that there is a possibility that the event will occur.
In general, probability has many excellent applications in games, commerce, and this newly growing area of artificial intelligence
The chance of an event can be calculated using the probability formula by only dividing the favourable number of possibilities by the total number of potential outcomes.
hence, DUM={GG,BB,GB,BG)
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The population of certain city is projected to grow at the rate of r(t) = 400 1+ people/ 24 +7 year in interval (Osts 5) t years from now. The current population is 60 000. What will be the population 5 years from now?
Answers
The population 5 years from now would be 60482
The population growth rate is given as:
\(r(t) = 400(1 + \frac{2t}{24 + t^2})\)
The value of t, 5 years from now is represented as:
t = 5
Substitute 5 for t in the function r(t).
So, we have:
\(r(5) = 400(1 + \frac{2 \times 5}{24 + 5^2})\)
Evaluate the exponent
\(r(5) = 400(1 + \frac{2 \times 5}{24 + 25})\)
Evaluate the products
\(r(5) = 400(1 + \frac{10}{24 + 25})\)
So, we have:
\(r(5) = 400(1 + \frac{10}{49})\)
Divide 10 by 49
\(r(5) = 400(1 + 0.204)\)
This gives
\(r(5) = 400(1.204)\)
Expand
\(r(5) = 481.6\)
The current population is given as 60000.
So, the population (P) 5 years from now would be
\(P = 60000 + 481.6\)
\(P = 60481.6\)
Approximate
\(P = 60482\)
Hence, the population (P) 5 years from now would be 60482
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i need the answer help
Answers
Answer:
the second one(20)
Step-by-step explanation:
If you pretend 20 is x, you solve the equation by pluging in x (20) into the equation and it gives you 70, 70 is what the big part is and 70+20 is 90, which is the size of a 90degree side.
4(20)-10
80-10
70
70+20=90
Answer: The answer to this question is 20 degrees
Step-by-step explanation:
So, you know that it's a straight line, which is 180 degrees. The angle on the very left is 90 degrees because it's a right angle.
So that means (x + 10) + x = 90 degrees because 180 - 90 = 90
Solve that equation for x:
x + 4x - 10 = 90
5x = 100
x = 20
So, your final answer is 20 degrees
A rectangle has an area of 24 square yards.the dimensions are whole numbers what are all the possible dimensions of the rectangle
Answers
Answer:
4 combination if order doesn't matter, 8 if order matter (for example is not the same 1x24 or 24x1)
Step-by-step explanation:
1x24
2x12
3x8
4x6
IF order matter also:
6x4
8x3
12x2
24x1
A turtle can walk 2/24 of a kilometer in an hour. The turtie is 2/10 of a kilometer away
from a pond. At this speed, how long will it take the turtle to reach the pond?
2 2/5hours
1 3/5hours
2 hours
3 1/5hours
Answers
Round0.00359 to nearest ten thousand
Answers
Answer: 0.0036
Step-by-step explanation:
0.00359 — 0.0036
The nine rounds up
Construct the 99% confidence interval estimate of the mean wake time for a population with the treatment
m
(Round to one decimal place as needed)
ample Get more help-
HW Score: 39.53%, 17 of 43 points
O Points: 0 of 6
A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 14 subjects had a mean wake time of 105 0 min After treatment, the 14 subjects had a
mean wake time of 782 min and a standard deviation of 24 1 min Assume that the 14 sample values appear to be from a normally distributed population and construct a 99% confidence interval estimate of the
mean wake time for a population with drug treatments What does the result suggest about the mean wake time of 105 0 min before the treatment? Does the drug appear to be effective?
Answers
The result suggests that the mean wake time might have really reduced since the values barely fall above 100 min as in before treatment with a high degree of confidence. thus , the drug is effective.
Confidence interval is written in the form as;
(Sample mean - margin of error, sample mean + margin of error)
The sample mean represent x , it is the point estimate for the population mean.
Margin of error = z × s/√n
Where s = sample standard deviation = 21.8
n = number of samples = 17
Now the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score
then the degree of freedom, df for the sample.
df = n - 1 = 17 - 1 = 16
Since confidence level = 99% = 0.99, α = 1 - CL = 1 – 0.99 = 0.01
α/2 = 0.01/2 = 0.005
Therefore the area to the right of z0.005 is 0.005 and the area to the left of z0.005 is 1 - 0.005 = 0.995
the t distribution table, z = 2.921
Margin of error = 2.921 × 21.8/√17
= 15.44
The confidence interval for the mean wake time for a population with drug treatments will be; 90.3 ± 15.44
The upper limit is 90.3 + 15.44 = 105.74 mins
The lower limit is 90.3 - 15.44 = 74.86 mins
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What are the coordinates of the verticals of the image of triangle DEF at the dilation Sincer and Fe with a scale factor of one over three followed by a translation along Victor -3,5
Answers
The rule applied to obtain each vertex of the triangle DEF after the transformations is given as follows:
(x,y) -> 1/3(x - 3, y + 5).
How to obtain the coordinates of each vertex?The original coordinates of triangle DEF have the following format:
(x,y).
The dilation by a scale factor of 1/3 means that each of the coordinates is multiplied by 1/3, hence:
(x,y) -> 1/3(x,y).
The translation along the vector <-3,5> means that the x-coordinate is subtracted by 3 while 5 is added to the y-coordinate, hence:
(x,y) -> 1/3(x - 3, y + 5).
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HELP PLEASE!!
Quadrilateral CDEF is a rhombus. What is m
Answers
Answer:
∠ BDC = 29°
Step-by-step explanation:
the sides of a rhombus are congruent, so CD = ED and Δ EDC is therefore isosceles with base angles congruent , then
∠ BCD = ∠ BED = 61°
• the diagonals are perpendicular bisectors of each other , then
∠ CBD = 90°
the sum of the 3 angles in Δ BCD = 180°
∠ BDC + ∠ CBD + ∠ BCD = 180°
∠ BDC + 90° + 61° = 180°
∠ BDC + 151° = 180° ( subtract 151° from both sides )
∠ BDC = 29°
Which expression is equivalent to 20 + 5 X 9
Answers
Which function is decreasing on the same interval as the function graphed here?
f (x)
61
N
-2
O A.
O B.
O c.
O D
4
2
-2
10
-41
2
A.
★
k (2) = -2x2 – 82 + 5
भ
Ko
j (±) = 22 + 4 – 4
=
9 (‡) = 322
- 12x + 18
h (x) = 2x2 + 8x + 3
Answers
The function that is decreasing on the same interval as the given function f(x) is h(x) = 2x^2 + 8x + 3.
To determine if a function is increasing or decreasing, we look at the sign of its derivative. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.
Taking the derivative of h(x) with respect to x, we get h'(x) = 4x + 8. To find the interval on which h(x) is decreasing, we need to find the values of x for which h'(x) < 0.
Setting h'(x) < 0, we have 4x + 8 < 0. Solving this inequality, we find x < -2.
Therefore, h(x) is decreasing for x < -2. Since the interval where h(x) is decreasing matches the interval for the given function f(x), we can conclude that h(x) = 2x^2 + 8x + 3 is the function that is decreasing on the same interval as f(x).
Overall, the function h(x) = 2x^2 + 8x + 3 is decreasing on the interval x < -2, which aligns with the interval of the given function f(x).
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(-4.2)²×..........=-132.3
Answers
Answer:
-7.5
Step-by-step explanation:
-4.2×-4.2 = 17.64
-132.3/17.64= -7.5
-16 + m = -6 what is the value of m
quickly please I'm on a limit
Answers
Your final answer should be : m=10
it takes and airplane, flying with the wind, 4 hours to reach boston from denver. when an airplane flew back against the wind it took 5 hours to reach denver from boston.what is the speed of the airplane if the speed of the wind is 50 mph
Answers
========================================================
Explanation:
a = speed of airplane without any wind
a+50 = speed of the plane with the wind boosting it
distance = rate*time
d = r*t
d = (a+50)*4
This represents the plane traveling 4 hours at the speed of a+50 mph. We don't really care about the distance d.
The other equation to set up is
d = (a-50)*5
which means the plane is going slower at a-50 mph and travels for 5 hours this time.
The system of equations is
\(\begin{cases}d = (a+50)*4\\d = (a-50)*5\end{cases}\)
Both equations involve the same variables 'a' and d, which allows us to use substitution to solve for 'a' like so
d = (a+50)*4
(a-50)*5 = (a+50)*4
5a-250 = 4a+200
5a-4a = 200+250
a = 450
The speed of the plane, without any wind, is 450 mph
Please look at the photo for the question. Thank you!
Answers
The zeros with each multiplicity are given as follows:
Multiplicity one: x = 6.Multiplicity two: x = 11.Multiplicity three: x = -6 and x = -5.How to obtain the multiplicities?The factor theorem is used to define the functions, which states that the function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.
Considering the linear factors of the function in this problem, the zeros are given as follows:
(x + 6)³ -> zero at x = -6 with multiplicity of 3.(x - 11)² -> zero at x = 11 with multiplicity of 2.x - 6 -> zero at x = 6 with multiplicity of 1.(x + 5)³ -> zero at x = -5 with multiplicity of 3.More can be learned about the Factor Theorem at brainly.com/question/24729294
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James wants to have earned $6,180 amount of interest in 28 years. Currently he finds
that his annual interest rate is 6.12%. Calculate how much money James needs to invest
as his principal in order to achieve this goal.
Answers
Answer:
$3606.44
Step-by-step explanation:
The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.
To do this, we need to use the formula for simple interest:
\(\boxed{I = \frac{P \times R \times T}{100}}\),
where:
I = interest earned
P = principal invested
R = annual interest rate
T = time
By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:
\(6180 = \frac{P \times 6.12 \times 28}{100}\)
⇒ \(6180 \times 100 = P \times 171.36\) [Multiplying both sides by 100]
⇒ \(P = \frac{6180 \times 100}{171.36}\) [Dividing both sides of the equation by 171.36]
⇒ \(P = \bf 3606.44\)
Therefore, James needs to invest $3606.44.
Need help with the provided questions
Answers
If Audrey wanted to solve this equation using the square root method, which answer shows the correct sequence of steps?
x2 + 5 = 41
a. Take the square root of both sides
b. Subtract 41 from both sides, then square root both sides.
c. Subtract 5 from both sides, then square root both sides.
d. Add 5 to both sides, then square root both sides
Answers
Answer:
C
Step-by-step explanation:
subtract 5 from both sides
I need an answer to this asap, hope you can understand.
Answers
The lengths of the segments x and y in the right triangles are x = 6√2 and y = 12√2
How to determine the lengths of x and yThe given shape is the right-angled triangle
Such that
We have angles = 30 degrees and 45 degrees
The measure of y can be calculated using the following sine ratio
sin(45) = y/24
Make y the subject
So, we have
y = 24 * sin(45)
When evaluated, we have
y = 12√2
The length x is calculated as
sin(30) = x/y
So, we have
x = y * sin(30)
This gives
x = 12√2 * 1/2
Evaluate
x = 6√2
Hence, the value of x in the triangle is 6√2
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Keisha, Miguel, and Ryan sent a total of 103 text messages during the weekend. Ryan sent 3 times as many messages as Miguel. Keisha sent 8 more
messages than Miguel. How many messages did they each send?
Number of text messages Keisha sent:
Number of text messages Miguel sent:
Number of text messages Ryan sent:
Answers
Answer:
only god knows
Step-by-step explanation:
because they didn't give us an answer on how many text messages anyone sent
What is 50% of 20?.........................
Answers
Answer:
10
Step-by-step explanation:
50%, or 1/2 of 20 is 10.
Answer:
10!
Step-by-step explanation:
50% is half, so half of 20 is 10!
which equation represents the slope intercept form of the line when the y intercept is (0,-6) and the slope is -5
Answers
The values into the slope-intercept form, we have y = -5x - 6
The slope-intercept form of a linear equation is given by:
y = mx + b
where 'm' represents the slope of the line, and 'b' represents the y-intercept.
In this case, the y-intercept is (0, -6), which means that the line crosses the y-axis at the point (0, -6).
The slope is given as -5.
Therefore, substituting the values into the slope-intercept form, we have:
y = -5x - 6
This equation represents the line with a y-intercept of (0, -6) and a slope of -5.
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A rectangular poster is 50 centimeters long and 25 centimeters wide. If 1 centimeter is approximately 0.4 inches, which of the following best represents the area of the poster in inches?
Answers
The area of the poster in inches is,
⇒ A = 4 inches²
We have to given that;
A rectangular poster is 50 centimeters long and 25 centimeters wide.
Here, 1 centimeter is approximately 0.4 inches
Hence, Lenght = 50 cm
Lenght = 50 x 0.4
= 2 inches
Width = 25 cm
= 25 x 0.4
= 1 inches
Thus, The area of the poster in inches is,
⇒ A = 1 x 4
⇒ A = 4 inches²
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carey correctly graphs a liner function. the slope of the function is -1. the y-intercept is 3. which is careys graph
Answers
Carey's graph will be a straight line with a slope of -1 and a y-intercept of 3.
Carey correctly graphs a linear function with a slope of -1 and a y-intercept of 3. A linear function is represented by the equation y = mx + b, where m is the slope and b is the y-intercept.
In this case, the slope is -1, which means that for every unit increase in the x-coordinate, the y-coordinate decreases by 1 unit. The y-intercept is 3, indicating that the graph intersects the y-axis at the point (0, 3).
To plot the graph, Carey starts by marking the point (0, 3) on the graph. Then, for every unit increase in the x-coordinate, Carey moves one unit downward. Similarly, for every unit decrease in the x-coordinate, Carey moves one unit upward. These steps ensure the correct slope of -1.
After connecting the points, Carey will obtain a line that starts at the y-intercept (0, 3) and slants downward, with a slope of -1. The resulting graph will be a straight line extending to both sides of the coordinate plane.
In conclusion, it is represented by the equation y = -x + 3.
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chapter 8 HELPPPPPPPPPP!!!!!!!!!!!!!!!!!
Find the circumference of each circle. Use your calculator's value of [pi]. Round your answer to the nearest tenth.
Answers
Answer:37.68/18.84
Step-by-step explanation:.
area of a circle s pier^2 or just 1/2pie r^2 so 1/2*3.148*2answer is 37.68.What are the two decisions that you can make from performing a hypothesis test? Select all that apply.
A.
accept the alternative hypothesis
B.
fail to reject the alternative hypothesis
C.
accept the null hypothesis
D.
fail to reject the null hypothesis
E.
reject the null hypothesis
F.
reject the alternative hypothesis
G.
make a type II error
H.
make a type I error
Answers
Answer:
reject the null hypothesis or fail to reject the null hypothesis.
The two decisions that perform the hypothesis test includes fail to deny the null hypothesis, and deny the null hypothesis.
Hypothesis testing refers to the testing that can give access to the reliability of the hypothesis via sample data.
There are two types of decisions that are considered to be wrong while performing the hypothesis test is:
In the case when the hypothesis is true but i.e. rejected. This is known as a type I error.In the case when the hypothesis is false, but it is failed to reject it. This is known as a type II error.So, the decisions that should not be considered while performing the hypothesis test is:
Acceptance of the alternative hypothesis.Acceptance of the null hypothesis.Fail to deny the alternative hypothesis.Deny the alternative hypothesis.Making of type I error.Making of type II error.Therefore we can conclude that options D and option E are considered from performing a hypothesis test.
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Natalie invests $2,000 into a savings account
which earns 11% per year. In 20 years, how
much will Natalie's investment be worth if
interest is compounded monthly? Round to the
nearest dollar.
Answers
Answer:
We can use the formula for compound interest to find the future value (FV) of Natalie's investment:
FV = P * (1 + r/n)^(n*t)
Where:
P is the principal amount (the initial investment), which is $2,000 in this case
r is the annual interest rate as a decimal, which is 11% or 0.11
n is the number of times the interest is compounded per year, which is 12 since interest is compounded monthly
t is the number of years, which is 20
Substituting the values into the formula, we get:
FV =
2
,
000
∗
(
1
+
0.11
/
12
)
(
12
∗
20
)
�
�
=
2,000 * (1.00917)^240
FV = $18,255.74
Therefore, after 20 years of compounded monthly interest at a rate of 11%, Natalie's investment of 2,000 will be worth approximately 18,256.
Answer:
$17,870
Step-by-step explanation:
You must use the formula for compound interest.
A = P(1 + r/n)^nt
I suggest you look it up at some point so that you can do these more easily in the future!