Answer:
S7 = 175S17 = 2465Sn = 1/2(n³ +n)Step-by-step explanation:
The progression of sums is ...
1, 5, 15, 34, 65, ...
So, first differences are ...
4, 10, 19, 31
Second differences are ...
6, 9, 12, ...
Third differences are constant:
3, 3, ...
This means the expression for Sn will be a cubic expression. If dn is the first of the n-th differences, then the equation can be written as ...
Sn = S1 +(n -1)(d1 +(n -2)/2(d2 +(n -3)/3(d3)))
And this simplifies a little bit to ...
Sn = 1 +(n -1)(4 +(n -2)(n +3)/2)
In simpler form, we have ...
Sn = 1/2(n³ +n)
Then the two terms we're interested in are ...
S7 = (1/2)(7³ +7) = 175
S17 = (1/2)(17³ +17) = 2465
Each term Sₙ consists of the sum of a triangular number of terms, which are given by
[tex]T_n = \displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]
The triangular numbers are given recursively for n ≥ 1 by
[tex]T_n = T_{n-1} + n[/tex]
starting with T₀ = 0.
For example,
• S₁ = 1 and
[tex]\displaystyle S_1 = \sum_{k=T_0+1}^{T_1} k = \sum_{k=1}^1 k = 1[/tex]
• S₂ = 2 + 3 and
[tex]\displaystyle S_2 = \sum_{k=T_1+1}^{T_2} k = \sum_{k=2}^3 k = 2 + 3[/tex]
• S₃ = 4 + 5 + 6 and
[tex]\displaystyle S_3 = \sum_{k=T_2+1}^{T_3} k = \sum_{k=4}^6 k = 4 + 5 + 6[/tex]
Then the n-th term of the sequence we're considering is
[tex]S_n = \displaystyle \sum_{k=T_{n-1}+1}^{T_n} k = \sum_{k=T_{n-1}+1}^{T_{n-1}+n} k[/tex]
Expanding this sum, we have
[tex]S_n = \left(T_{n-1}+1\right) + \left(T_{n-1}+2\right) + \left(T_{n-1}+3\right) + \cdots + \left(T_{n-1}+n\right)[/tex]
There are n terms on the right side, and hence n copies of [tex]T_{n-1}[/tex], and the rest of the terms make up the next triangular number [tex]T_n[/tex] :
[tex]S_n = nT_{n-1} + 1 + 2 + 3 + \cdots + n[/tex]
[tex]S_n = nT_{n-1} + \displaystyle \sum_{k=1}^n k[/tex]
[tex]S_n = nT_{n-1} + T_n[/tex]
We have a closed form for [tex]T_n[/tex], so we end up with
[tex]S_n = n \cdot \dfrac{(n-1)n}2 + \dfrac{n(n+1)}2 \implies \boxed{S_n=\dfrac{n^3+n}2}[/tex]
From here it's easy to find S₇ and S₁₇.
[tex]S_7 = \dfrac{7^3+7}2 \implies \boxed{S_7 = 175}[/tex]
[tex]S_{17} = \dfrac{17^3+17}2 \implies \boxed{S_{17} = 2465}[/tex]
A car traveled 133 miles in 4 3/4 hours. What was the cars average speed in miles per hour?
4 3/4 = 19/4
133/(19/4) = 133(4/19) = 1197/19 = 63 mph
12POINTS PLEASE HURRY ASAPPPPP <3
ITS SQAURE 6
6 SQAURE OFTHE 6
Choose the fraction that goes in the blank. 5/12< _____<3/4
Answer:
7/12 or 8/12 are both good options
Step-by-step explanation:
nvr mind, its 7/12 only
Please due today by 10:00
The length of my rectangular calculator lid is 11 mm more than the width. The area is 152 square
mm. Find the dimensions of the lid.
Answer:
The dimensions of the lid are 8mm by 19mm.
w = 8
l = 19
Step-by-step explanation:
[tex]l=w+11\\l\times w=152[/tex]
where l is length and w is width. This can be solved as a system of equations.
[tex]l\times w=152\\(w+11)\times w=152\\w(w+11)=152\\w^2+11w=152\\w^2+11w-152=0[/tex]
At this point, it gets a little tough. I might be unnecessarily overcomplicating things, but this is the only way I see to solve the problem.
=================== Skip down below if you don't care about factoring
You need to factor the newly created trinomial.
[tex]ax^2+bx+c[/tex]
With a trinomial in this form, you need to find 2 numbers that add together to make b and multiply together to make ac.
Here, we need 2 numbers that add to 11 and multiply to -152. First, factor 152:
1, 152
2, 76
4, 38
8, 19
Then the reverse of all of those is true too, of course:
19, 8
38, 4
etc
In our case, we're looking for -152, so one of our factors will be negative. We're also looking for factors that add up to 11. Looking at these factors, you can see that 19 - 8 = 11, so our factors are 19 and -8.
Finally, you can use those to factor our trinomial. Split up the middle number (11w) into two:
[tex]w^2+11w-152=0\\w^2-8w+19w-152=0[/tex]
And now, you can factor by grouping:
[tex]w^2-8w+19w-152=0\\w(w-8)+19(w-8)=0\\(w+19)(w-8)=0[/tex]
===================
Now that the number is factored, you can finally find w:
[tex](w+19)(w-8)=0[/tex]
Here, you can see that the equation will be true when w = -19 or w = 8. Those are our solutions, but we can't have a negative distance, so it's just
[tex]w=8[/tex]
Going all the way back to the top, now you can use the width to find the length.
[tex]l=w+11\\l=8+11\\l=19[/tex]
That one was much easier.
The dimensions of the lid are 8mm by 19mm.
Finally, check that with both of the original equations to make sure it's correct.
[tex]l=w+11\\19=8+11\\19=19\\\\l\times w=152\\19\times8=152\\152=152[/tex]
After deducting grants based on need, the average cost to attend the University of Southern California (USC) is $27,175 (U.S. News & World Report, America’s Best Colleges, 2009 ed.). Assume the population standard deviation is $7,400. Suppose that a random sample of 60 USC students will be taken from this population.
a. Refer to Exhibit 2. What is the value of the standard deviation of the mean? (Note: keep two decimal places.)
b.What is the probability that the sample mean will be more than $27,175
c.What is the probability that the sample mean will be within $1,000 of population mean? (Note: keep two decimal places for the z value and four decimal places for the final probability value.)
d.What would be the probability in Part c if the sample size were increased to 100? (Note: keep two decimal places for the z value and four decimal places for the final probability value.)
The solutions are:
(a) Standard error is 957.04. (b) The probability is 50%. (c)The probability is 0.8491. (d) The probability is 0.8617.
To calculate the values, use the standard error formula for the sample mean:
(a)
The standard error is given by:
[tex]Standard\ error =\dfrac{ Population \ standard \ deviation}{\sqrt{sample\ size}}\\Standard\ error=\dfrac {\$ 7,400 }{ \sqrt60}\\Standard\ error =\ $957.04[/tex]
(b)
To find the probability that the sample mean will be more than $27,175, convert it into a z-score and then find the area under the normal distribution curve:
z = (sample mean - population mean) / standard error
z = ($27,175 - $27,175) / $957.04
z = 0
Since the z-score is 0, the area under the curve to the right of this point is 0.5 or 50%.
c. To find the probability that the sample mean will be within $1,000 of the population mean, we need to find the area between the range of $(27,175 - $1,000) and $(27,175 + $1,000).
Lower limit: $27,175 - $1,000 = $26,175
Upper limit: $27,175 + $1,000 = $28,175
Now, we calculate the z-scores for both limits:
lower = ($26,175 - $27,175) / $957.04 ≈ -1.045
upper = ($28,175 - $27,175) / $957.04 ≈ 1.045
Using a standard normal distribution table or calculator, the probability of a z-score being between -1.045 and 1.045 is approximately 0.8491.
(d)
If the sample size were increased to 100.
[tex]Standard error = \dfrac{ \$7,400}{ \sqrt{100}} \\Standard error = \$740[/tex]
Find the probability that the sample mean will be within $1,000 of the population mean, did in part c:
Lower limit: $27,175 - $1,000 = $26,175
Upper limit: $27,175 + $1,000 = $28,175
lower = ($26,175 - $27,175) / $740
lower ≈ -1.351
upper = ($28,175 - $27,175) / $740
upper ≈ 1.351
Using a standard normal distribution table or calculator, the probability of a z-score being between -1.351 and 1.351 is approximately 0.8617.
Hence,
Standard error is 957.04.The probability is 50%.The probability is 0.8491.The probability is 0.8617.Learn more about Probability here:
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help me please!! <33
13/20 - 1/2
===================================================
Explanation:
Ignore the big X and the shading for now. This figure has 4 rows by 5 columns of rectangles. So there are 4*5 = 20 little rectangles total. Each little rectangle represents 1 brownie.
Then let's say the shaded region represents the amount of brownies Mary receives. We have 13 rectangles shaded in out of the 20 total. This means 13/20 is the fractional amount Mary gets.
She then eats 10 of the 20 total, so she eats 10/20 = 1/2 of the brownies.
Therefore, we end up with the expression 13/20 - 1/2 to represent the fractional amount that Mary still has that she hasn't eaten yet (which is 3/20).
We can say: 13/20 - 1/2 = 13/20 - 10/20 = 3/20
Am i blind or does this just not make any sense? Pls help!
Answer:
0.4
Step-by-step explanation:
2/5 converted into hundredths is 40/100.
40/100 as a decimal is 0.40, or 0.4
-hope it helps
wren is going to add an ordered pair to the table that will cause the table to no longer be a function. Which could be the ordered pair?
X Y
-3 7
-1 5
0 3
2 6
_ _
Answer:
(0,-2)
Step-by-step explanation:
What is an equation of the line that passes through the points (-4, -5) and (-2, -6)?
Answer: [tex]y=-\frac{1}{2}x-7[/tex]
Ok done. Thank to me :>
if the length of a cuboid is 60cm and width 40cm and its surface area is 14800cm² find its height
Please Guyz I Want Full Solved Not Just Answer Please
Answer:
The height of cuboid is 50 cm.
Step-by-step explanation:
Given :
➺ Length of cuboid = 60cm➺ Width of cuboid = 40cm.➺ Surface area of cuboid = 14800cm²To Find :
➺ Height of cuboidUsing Formulas :
[tex]\star\small{\underline{\boxed{\sf \pink{SA_{(Cuboid)} = 2(lb + bh + lh)}}}}[/tex]
»» SA = surface area »» l = length »» b = breadth »» h = heightSolution :
Substituting all the given values in the formula to find height of cuboid :
[tex]{\longrightarrow{\sf{SA_{(Cuboid)} = 2(lb + bh + lh)}}}[/tex]
[tex]{\longrightarrow{\sf{14800= 2(60 \times 40 + 40 \times h + 60 \times h)}}}[/tex]
[tex]{\longrightarrow{\sf{14800= 2(2400 + 40h + 60h)}}}[/tex]
[tex]{\longrightarrow{\sf{14800= 2(2400 + 100h)}}}[/tex]
[tex]{\longrightarrow{\sf{14800= 4800 + 200h}}}[/tex]
[tex]{\longrightarrow{\sf{200h = 14800 - 4800}}}[/tex]
[tex]{\longrightarrow{\sf{200h = 10000}}}[/tex]
[tex]{\longrightarrow{\sf{h = \dfrac{10000}{200}}}}[/tex]
[tex]{\longrightarrow{\sf{h = \cancel{\dfrac{10000}{200}}}}}[/tex]
[tex]{\longrightarrow{\sf{\underline{\underline{\red{h = 50 \: cm}}}}}}[/tex]
Hence, the height of cuboid is 50 cm.
[tex]\underline{\rule{220pt}{3.5pt}}[/tex]
A rectangular cube of iron is a side 8 cm. What is it volume.
[tex] = {8}^{3} \: \: cm ^{3} \\ = 512 \: \: {cm}^{3} [/tex]
Answer:
The volume is 512 cube centimetres.
Hope you could get an idea from here.
Doubt clarification - use comment section.
PLZZZ HELP, BEST ANSWER GETS BRANLIEST AND ITS DUE IN 5 MINUTESSS,
The figure shows two triangles on the coordinate grid:
Which of the following statements is true about the three quadrilaterals? (5 points)
A and B are similar but not congruent.
A and C are similar but not congruent.
A and B are similar and also congruent.
B and C are similar and also congruent.
Answer:
The first one
Step-by-step explanation:
A and B are similar because the proportions are the same.
A is congruent to C but none of the answers ask that.
help plsssssssssssssssssssssssssssss
Answer:
0
the answer is 0........
USE THE DIATRIBUTIVE PROPERTY TO WRITE 6(W+5) AS AN EQUIVALENT EXPRESSION
Answer:
w = 5
Step-by-step explanation:
Step 1: Use the Distributive property to get 6w + 30.
Step 2: Divide 6w by 6 and cancel out both 6s.
Step 3: Divide 30 by 6 to get 5, and you have the answer.
y = 30.078(1.003)*.
Question 1
? Question
Use the exponential model to make predictions about closing prices of this stock. Remember that x
represents the number of weeks since the first data point, so the value of x today would be 260.
Type the correct answer in each box. If necessary, round the answers to the nearest cent.
One week from today, the stock is predicted to close at $
One month from today, the stock is predicted to close at $
One year from today, the stock is predicted to close at $
Using the exponential function, it is found that:
One week from today, the stock is predicted to close at $65,734.37.One month from today, the stock is predicted to close at $66,327.75.One year from today, the stock is predicted to close at $76,584.35.The exponential function is:
[tex]y(x) = 30078(1.003)^x[/tex]
x represents the number of weeks since the first data point, so the value of x today would be 260.Hence, considering today as an starting point, the function is:
[tex]y(x) = 30078(1.003)^{x + 260}[/tex]
One week from now, we have y(1), hence:
[tex]y(1) = 30078(1.003)^{1 + 260} = 65734.37[/tex]
One week from today, the stock is predicted to close at $65,734.37.
One month from now, we have y(4), hence:
[tex]y(4) = 30078(1.003)^{4 + 260} = 66327.75[/tex]
One month from today, the stock is predicted to close at $66,327.75.
One year from now, we have y(52), hence:
[tex]y(52) = 30078(1.003)^{52 + 260} = 76584.35[/tex]
One year from today, the stock is predicted to close at $76,584.35.
To learn more about exponential functions, you can take a look at https://brainly.com/question/25958656
Answer:
One week from today, the stock is predicted to close at $65,734.37.
One month from today, the stock is predicted to close at $66,327.75.
One year from today, the stock is predicted to close at $76,584.35.
Step-by-step explanation:
Darla scored 18 points in her last basketball game, shooting only 2- and 3-point shots. If she made 7 total
Answer:
3 2-pt shots, 4 3-pt shots
Step-by-step explanation:
2x + 3y = 18
x + y = 7, so x = 7 - y
Substitute in for x
2(7 - y) + 3y = 18
14 - 2y + 3y = 18
14 + y = 18
y = 4
Now put 4 in for y to solve for x
2x + 3(4) = 18
2x + 12 = 18
2x = 6
x = 3
The camp cook needed 2172 ounces of juice each day each carton of juice contains 64 ounces the Cook estimated that he needed 30 cartons of juice each day which expression proves that his estimate is incorrect
Answer: 384
Step-by-step explanation:
URGENT simplify (5^3 x 5^5)^5
Answer:
5^40
Step-by-step explanation:
Step 1: Multiply the exponents within the parentheses with the exponent outside of the parentheses.
[tex](5^3^*^5)(5^5^*^5)[/tex] [tex](5^1^5)(5^2^5)[/tex]Step 2: Based on the laws of exponents, add the exponents together.
[tex]5^(^1^5^+^2^5^)[/tex] [tex]5^4^0[/tex]Therefore, the answer is 5^40.
A person of 600 newtons
gets on an elevator which
lifts him 6 meters in
10 seconds. How much
power was used?
Answer:
360J/s
Step-by-step explanation:
power=work done/ time
power=600×6/10
=3600/10
=360
In a survey of 1900 people who owned a certain type of car, 1140 said they would buy that type of car again. What percent of the people surveyed were satisfied with the car?
Answer:
60% of the people surveyed were satisfied with the car
Step-by-step explanation:
1140 satisfied people / 1900 total people = 0.60 = 60% satisfied people
[tex]f(x) = {x}^{3 } + {3x}^{2} - 9x + 5[/tex]
which of the following are roots of a polynomial function?
check all that apply
a[tex]1 - \sqrt{3} [/tex]
b
[tex]1 + \sqrt{3} [/tex]
c
[tex] - 5[/tex]
D
[tex]3 - \sqrt{2} [/tex]
E
[tex]3 + \sqrt{2} [/tex]
F
[tex]1[/tex]
Answer:
[tex]f(1) = {(1)}^{3} + 3 {(1)}^{2} - 9(1) + 5 \\ (x - 1) = 0 \rightarrow \: x = 1 \\\frac{{x}^{3 } + {3x}^{2} - 9x + 5}{x - 1} = {x}^{2} + 4x - 5 \\ (x + 5)(x - 1) = 0 \\ x = - 5 \: or \: x = 1 \\ \color{red}\boxed{x_{1} = 1} \\\color{yellow} \boxed {x_{2} = 1} \\ \color{green}\boxed{x_{3} = - 5} [/tex]
The answer would be C and F
Answer: 1 & -5
Step-by-step explanation
a.
two more than seven times x
b. the product of ten and f,
increased by one
C. r divided by eleven
Answer:
A. two more than seven times x7 × x +27x+2B. the product of ten and f increased by one10 × f + 110f+1C. r divided by elevenr ÷ 11Rewrite in simplest terms: (-9x + 7y) - (-9x + 5y)
Answer:
3y
Step-by-step explanation:
(-9x+7y)-(-9x+5y)
Distribute the negative(-)
-9x+7y+9x-5y
-9 and 9 cancel each other
3y
The simplest terms of the given expression (-9x + 7y) - (-9x + 5y) will be 2y.
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
A statement expressing the equality of two mathematical expressions is known as an equation.
As per the given expression,
(-9x + 7y) - (-9x + 5y)
Open the bracket
-9x + 7y + 9x - 5y
Combined likely terms together
-9x + 9x + 7y - 5y
0 + 2y = 2y
Hence "The formula (-9x + 7y) - (-9x + 5y) will have 2y as its simplest terms".
To learn more about expression,
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What are the measures of the other three angles formed by the intersection
Answer:
angle 1 130°
angle 2 50°
angle 3 130°
Step-by-step explanation:
180°-50°=130°, use one of the road as the straight line,
which is 180° so angle 1 is 130°
which means angle 2 is also 50° because 180°-130° is 50°
and angle 3 is 130° because 180°-50°=130°
Hope this helps :)
Use the GCF and the Distributive property to find the sum of 66+78
Here is ur answer of your question
There are two spheres; the radius of the big sphere is two times the radius of the small sphere. How many times larger is the volume of the big sphere compared to the small one?
The volume of the bigger sphere is 8 times larger than the volume of the smaller sphere.
The formula of the volume of a sphere is given below
⇒ Formula:
V = 4/3(πr³)................. Equation 1⇒ Where:
V = Volume of the spherer = radius of the sphereπ = constant called pieFrom the question, There are two spheres.
If the radius of the bigger sphere is two times the radius of the smaller sphere.
R = 2r⇒ Where
R = bigger radiusr = smaller radiusFor the bigger sphere
⇒ Substitute these values into equation 1
V = 4/3(π(2r)³)V = 4/3(π8r³)............. Equation 2For the smaller sphere,
V' = 4/3(πr³)............... Equation 3⇒ Comparing equation 2 and equation 3
V = 8V'Hence, The volume of the bigger sphere is 8 times larger than the volume of the smaller sphere.
Learn more about volume here: https://brainly.com/question/1972490
Factor: 3x4y3 – 48y3
Answer: 4y3(1x1 - 12)
3x4y3 – 48y3
4y3(1x1 - 12)
What is the value of x in the equation 8+4 = 2(x-1)2
O 5
11
o
2
끌
o
13
2
O 7
Step-by-step explanation:
→ 8 + 4 = 2(x -1)
→ 12 = 2x - 2
→ 2x - 2 = 12
→ 2x = 12 + 2 = 14
→ x = 14/2 = 7
→ x = 7
hence, option D (7) is correct.
Hope this answer helps you dear ! take care
Find the 60" term of the following arithmetic sequence.
6, 13, 20, 27,
..
[tex]\text{First term, a =6}\\\\\text{Common difference, d = 13 -6 = 7}\\\\\text{Nth term} = a+(n-1)d = 6 +(n-1)7 = 6+7n-7 = 7n-1\\\\\text{60th term} = 7(60) -1 = 419[/tex]