Answers
a. n =17
b. n =23
c. n = 9
d. n = 15
Answer:
(a) n = 17
(b) n = 23
(c) n = 9
(d) n = 15
Step-by-step explanation:
Sum of the first n terms of an arithmetic series:
[tex]\boxed{S_n=\dfrac{n}{2}(a+a_n)}[/tex]
where:
a is the first term.aₙ is the nth term.Part (a)[tex]\displaystyle \sum^n_{i=1}(3i+5)=544[/tex]
To find a substitute i = 1 into the equation:
[tex]\implies a=3(1)+5=8[/tex]
To find aₙ substitute i = n into the equation:
[tex]\implies a_n=3n+5[/tex]
Substitute the values of a, aₙ and Sₙ into the arithmetic series formula and solve for n:
[tex]\begin{aligned}S_n & =\dfrac{n}{2}(a+a_n)\\ \implies 544&=\dfrac{n}{2}(8+3n+5)\\1088&=n(3n+13)\\1088&=3n^2+13n\\3n^2+13n-1088&=0\\3n^2-51n+64n-1088&=0\\3n(n-17)+64(n-17)&=0\\(n-17)(3n+64)&=0\\\\n-17&=0 \implies n=17\\3n+64&=0 \implies n=-\dfrac{64}{3} \end{aligned}[/tex]
As n is positive, n = 17.
Part (b)[tex]\displaystyle \sum^n_{i=1}(-4i-1)=-1127[/tex]
To find a substitute i = 1 into the equation:
[tex]\implies a=-4(1)-1=-5[/tex]
To find aₙ substitute i = n into the equation:
[tex]\implies a_n=-4n-1[/tex]
Substitute the values of a, aₙ and Sₙ into the arithmetic series formula and solve for n:
[tex]\begin{aligned}S_n & =\dfrac{n}{2}(a+a_n)\\ \implies -1127&=\dfrac{n}{2}(-5-4n-1)\\-2254&=n(-4n-6)\\-2254&=-4n^2-6n\\4n^2+6n-2254&=0\\2n^2+3n-1127&=0\\2n^2-46n+49n-1127&=0\\2n(n-23)+49(n-23)&=0\\(2n+49)(n-23)&=0\\\\2n+49&=0 \implies n=-\dfrac{49}{2}\\n-23&=0 \implies n=23 \end{aligned}[/tex]
As n is positive, n = 23.
Part (c)[tex]\displaystyle \sum^n_{i=5}(7+12i)=455[/tex]
[tex]\boxed{\displaystyle \sum _{i=m}^n\:=\:\sum _{i=1}^n\:-\:\sum _{i=1}^{m-1}}[/tex]
Therefore:
[tex]\displaystyle \sum^n_{i=1}(7+12i)-\sum^{4}_{i=1}(7+12i)=455[/tex]
To find a substitute i = 1 into the equation:
[tex]\implies a=7+12(1)=19[/tex]
To find aₙ substitute i = n into the equation:
[tex]\implies a_n=7+12n[/tex]
To find a₄ substitute i = 4 into the equation:
[tex]\implies a=7+12(4)=55[/tex]
Therefore:
[tex]\begin{aligned}\displaystyle \sum^n_{i=1}(7+12i)-\sum^{4}_{i=1}(7+12i)&=455\\\dfrac{n}{2}(19+7+12n)-\dfrac{4}{2}(19+55)&=455\\\dfrac{n}{2}(12n+26)-148&=455\\ \dfrac{n}{2}(12n+26)&=603\\6n^2+13n&=603\\6n^2+13n-603&=0\\ 6n^2-54n+67n-603&=0\\6n(n-9)+67(n-9)&=0\\(6n+67)(n-9)&=0\\\\6n+67&=0 \implies n=-\dfrac{67}{6}\\n-9&=0 \implies n=9\end{aligned}[/tex]
As n is positive, n = 9.
Part (d)[tex]\displaystyle \sum^n_{i=3}(-3-4i)=-507[/tex]
[tex]\boxed{\displaystyle \sum _{i=m}^n\:=\:\sum _{i=1}^n\:-\:\sum _{i=1}^{m-1}}[/tex]
Therefore:
[tex]\displaystyle \sum^n_{i=1}(-3-4i)-\sum^{2}_{i=1}(-3-4i)=-507[/tex]
To find a substitute i = 1 into the equation:
[tex]\implies a=-3-4(1)=-7[/tex]
To find aₙ substitute i = n into the equation:
[tex]\implies a_n=-3-4n[/tex]
To find a₂ substitute i = 2 into the equation:
[tex]\implies a=-3-4(2)=-11[/tex]
Therefore:
[tex]\begin{aligned}\displaystyle \sum^n_{i=1}(-3-4i)-\sum^{2}_{i=1}(-3-4i)&=-507\\\dfrac{n}{2}(-7-3-4n)-\dfrac{2}{2}(-7-11)&=-507\\\dfrac{n}{2}(-4n-10)+18&=-507\\-2n^2-5n&=-525\\2n^2+5n-525&=0\\ 2n^2-30n+35n-525&=0\\2n(n-15)+35(n-15)&=0\\(2n+35)(n-15)&=0\\\\2n+35&=0 \implies n=-\dfrac{35}{2}\\n-15&=0 \implies n=15 \end{aligned}[/tex]
As n is positive, n = 15.
Related Questions
Simplify c^3d^4/c^2d
Answers
Answer:
I think the answer is :-
[tex]c {d}^{3} [/tex]
The simplification of c^3d^4/c^2d is cd^3.
What is simplification of an expression?Simplification of an expression is the process of writing an expression in the most efficient and compact form without affecting the value of the original expression.
For the given situation,
The expression is c^3d^4/c^2d.
The simplification is as
⇒ [tex]\frac{c^{3}d^{4} }{c^{2}d }[/tex]
⇒ [tex]\frac{(c^{2})( c)(d^{3} )(d)}{c^{2} d}[/tex]
⇒ [tex]cd^{3}[/tex]
Hence we can conclude that the simplification of c^3d^4/c^2d is cd^3.
Learn more about simplification of an expression here
https://brainly.com/question/13730519
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Find the measure of angle x in the figure below:
BRAINLIEST PLEASE HELP SHOW WORK AN WILL MAKE BRAINLY
Answers
Answer:
See attached photo!
Step-by-step explanation:
write each improper fraction as a mix numbet 19/4
Answers
Answer:
4 3/4
Step-by-step explanation:
Describe what a graph of xs 1 would look like. (Be sure to include whether dot is open or closed and state
which way the arrow would point).
(Type answer here)
If i get it correct i give u
Answers
Answer:
The graph would be a line starting at one and moving to the left infinitely. The dot would be closed.
Please help very urgent
Answers
Answer:
1) A={0,2,4,12,14,20}
2) 6
3) W={1,3,5,9,21,23,39}
4) 7
The constraints of a problem are listed below. What are the vertices of the feasible region?
X+y<_5
y<_3
X>_0
y>_0
Answers
Answer:
(0, 0), (0, 3), (2, 3), and (5, 0)
Step-by-step explanation:
The constraints of the problem are:
x + y ≤ 5
y ≤ 3
x ≥ 0
y ≥ 0
Then:
Using the second and fourth inequalities we can write:
0 ≤ y ≤ 3
Knowing that the minimum value of y is 0, then if we replace that in the first equation we get:
x + 0 ≤ 5
x ≤ 5
While for the maximum value of y, y = 3, this inequality becomes:
x + 3 ≤ 5
x ≤ 2
Now, the vertexes are the points where both variables take one of their extremes.
Then, we have a vertex at (0, 0) because we have:
x ≥ 0
y ≥ 0
So this is the vertex when both x and y take the minimum value.
When y takes the maximum value y = 3, and x takes the minimum value x = 0, we have the vertex:
(0, 3)
When y takes the maximum value, y = 3, and x takes the maximum value, x = 2, we have the vertex:
(2, 3)
When y takes the minimum value, y = 0, and x takes the maximum value, x = 5, we have the vertex:
(5, 0)
Then the four vertexes are:
(0, 0), (0, 3), (2, 3), and (5, 0)
What is the completely factored form of the following expression
6b2+5b-4
Answers
What is the perimeter of the parallelograrn?
Answers
Please help on number one I’ve been so stuck
Answers
Answer/Step-by-step explanation:
Ok so x,y. is the first thing you need to know A. 1, -5 = X, -Y.
the number from the x grid goes before the y see the arrows seperating the squares the vertical line is the X any numbers coming off of x go before y ok, x, y.
See the -5, -5 on the bottom right square both of those are negative and since E. is negative (-2, -2) it gets placed in the negative bottom right square ok.
now you see how for some squares there is only 1 number for them well the number on the other side where there is no number is the number. In the top right square there is only 5 on the y line meaning that if u look beyond that and where the other number suppose to be the -5 is there so -5 is the answer for that ok.
(The 0 in the middle is called the "Origin")
Hope this helps! I didn't exactly answer because there's no way to answer without downloading or submitting an image but it wont let me download it for some reason.
Top right square = -5, 5.
C. goes into the top right square
Top left = 5, 5.
B. goes into the top left square
Bottom right square = -5, -5.
E. gets placed in the bottom right square
Bottom left square = 5, -5.
D. gets placed in the bottom left square
A. gets placed in the bottom left square
~ LadyBrain
|x+4|=6 please help but give me an answer other than 2
Answers
Answer: -10 (or 2)
Step-by-step explanation: Absolute value gets rid of the negative so you can solve it with a positive or negative number as long as the interger still has the same value. (|-2|=2 and |+2|=2) so |-10+4|=6
A total of 494 tickets were sold for the school play. They were either adult tickets or student tickets. There were 56 fewer student tickets sold than adult tickets. How many adult tickets were sold?
Answers
494=x+x-56
494=2x-56
550=2x
275=x
The number of adult tickets sold will be 275.
What is a mathematical function, equation and expression?function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators.equation : A mathematical equation is used to equate two expressions.Given is that a total of 494 tickets were sold for the school play. They were either adult tickets or student tickets. There were 56 fewer student tickets sold than adult tickets.
Assume that the number of adult tickets sold are [x] and the student tickets sold are [y]. Then -
x + y = 494
y = x - 56
So, we can write the equation as -
x + x - 56 = 494
2x = 550
x = 275
Then, we can find [y] as -
y = 275 - 56 = 219
y = 219
Therefore, the number of adult tickets sold will be 275.
To solve more questions on functions, expressions and polynomials, visit the link below -
brainly.com/question/17421223
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3.) A 50ft by 40ft one-story house sits in the middle of a
lot that is 80 ft by 60ft. If the edge of the house to the
edge of the lot is covered in grass, how many square feet
of grass will the owner mow?
Answers
Answer:
2800 ft^2
Step-by-step explanation:
NEED HELP!
A water tank at a filtration plant is built in the shape of an inverted cone with height 5.2
m and diameter 5 m at the top. Water is being pumped into the tank at a rate of 1.2 m3
/min.
Find the rate at which the water level is rising when there is 8π m3
of water
Answers
Answer:
The rate of change is
6.2 × [tex]10^{-3}[/tex] [tex]ms^{-1}[/tex] = 0.37 [tex]\frac{m}{min}[/tex]
Answer ^ ^ ^ ^ ^ ^ ^
Explanation: it broke itself
The table slow shows Larry's scores on 6 different math tests. What is Larry's mean score on the 6 tests?
Test: 1 2 3 4 5 6
Score: 88 78 76 86 92 96
Answers
Answer:
86
Step-by-step explanation:
Help me out I’m not sure what the answer is
Answers
The area (A) of a trapezoid is calculated as follows:
A = 1/2 * sum of the parallel sides * height
In this case:
A = 1/2 * (3.4 + 1.1) * 2 = 4.5 cm^2.
Provide the remaining principal parts for the verb below.
Present: shake
Present Participle:
Past:
Past Participle:
Answers
Past : Shook
Past Participle: Shaken
Factor the algebraic expression (x - 1)2
- (y - 2)2
Answers
Answer:
2(x-y+1) Is The Awnser
6. A vertex is a point where _____ or more edges meet.
Answers
Answer:
2
Step-by-step explanation:
greg made the dart board shown.it is divided into equal sedtions
Answers
Answer:
the answer is 20 percent
Answer:
the answer is 20 percent
PLEASE HELP ME, LIKE ACTUALLY! The sum of two numbers is 193. One of them in 71 more than the other. Find the larger of these two numbers
Answers
Answer:
132
Step-by-step explanation:
X is smaller number
x + 71 is larger number.
x+ (x+71)=193
2x = 193-71
2x=122
x=61
x+71=largest number
61+71=132 largest numher
check answer
x+(x+71)=193
61+132=193
Answer:
The larger of the two numbers is 132.
Step-by-step explanation:
let x be the larger number and y be the smaller number.
x + y = 193 [Equatio#1]
x = y +71 [Equation#2]
substitute x from equation #2 into equation # 1:
(y+71) + y = 193
2y = 193 - 71
2y = 122
y = 61
substitute this value of y back into equation #2 to solve for x:
x = (61)+71
x= 132
the larger number is 132.
Find the height and slant height of the cone. Round your answers to the nearest whole number.
Answers
Answer:
Height = 4 cm and slant height = 5 cm
Step-by-step explanation:
Given that,
The surface area of the cone, A = 75.4 cm²
Radius, r = 3 cm
The surface area of the cone is given by the formula as follows :
[tex]A=\pi r^2+\pi rl[/tex]
Put all the values,
[tex]75.4=3.14\times 3^2+3.14\times 3\times l\\\\75.4=28.26+9.42l\\\\75.4-28.26=9.42l\\\\47.14=9.42l\\\\l=\dfrac{47.14}{9.42}\\\\l=5\ cm[/tex]
Also,
[tex]l^2=r^2+h^2\\\\h=\sqrt{l^2-r^2}\\\\h=\sqrt{5^2-3^2}\\\\h=4\ cm[/tex]
Hence, the height of the cone is 4 cm and the slant height is 5 cm.
5. A parallelogram has vertices D(-1,1) E
(2,4) F(5. 1 )G(2,-2) Which description best describes the figure? A rectangle B trapezoid, C. Square, D. Kite
Answers
Answer:
C
Step-by-step explanation:
2a = 8
4b = 0
5c = 35
x/2 = 8
Answers
B=0
C=7
X=16
This is what I think
a=4
b=0/4
b=0
C=35/5
c=
LCM is 2
Therefore 2x=4
Then x=2
Write an inequality for the sentence: The maximum height his 45 feet.
Answers
Answer:
h ≤ 45
Step-by-step explanation:
The maximum height is 45 feets
Let the height of the object = h
Hence, the sentence means that the height, h of the object cannot be beyond 45 feets ;
This means the height can be any number of feets less than or equal to 45 feets
Hence, h ≤ 45
On the image, if you scroll, there are questions and answers. I just need one example on how to get to those answers. I’m not sure how to do these types of problems. If you can help I’d appreciate it! Thank you!!
Answers
Answer:
Step-by-step explanation:
#27). 6k² = 48
k = √8 = 2√2
x = [tex]\frac{8}{2\sqrt{2} }[/tex] = 2√2 cm
y = [tex]\frac{12}{2\sqrt{2} }[/tex] = 3√2 cm
#29). 16k² = 48
k = √3
x = [tex]\frac{8}{\sqrt{3} }[/tex] = [tex]\frac{8\sqrt{3} }{3}[/tex] cm
y = [tex]\frac{12}{\sqrt{3} }[/tex] = 4√3 cm
17. The present age of a father and his son is 50yrs and 20yrs respectively. How many years ago
the product of their ages was 451.
Answers
Answer:
9 yrs ago
Step-by-step explanation:
(50-x)(20-x) = 451
x^2-70x + 1000 = 451
x^2-70x + 549 = 0
(x-9)(x-61) = 0
x = 9 (if it was 61 yrs ago, the father and son both wouldnt be a live)
PLEASE PLEASE PLEASE HELP What is the measure of
T
175°
H
57°
F
Answers
Answer:
m<G = 59°
Step-by-step explanation:
Recall: the angle formed outside a circle when 2 secants intercept is equal to half of the difference of the intercepted arcs
m<G = ½(arc TE - arc HF)
m<G = ½(175 - 57)
m<G = ½(118)
m<G = 59°
What is the perimeter and area of this????? Please help fast
Answers
Hi there!
[tex]\large\boxed{\text{ P = 31ft, A = 35ft}^2}[/tex]
Perimeter: Add up all the side lengths.
14 + 10 + 7 = 31 ft.
Area: Use the following formula:
A = 1/2(bh) where:
b = bass
h = height
Thus:
A = 1/2(10 · 7) = 35 ft²
A quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard deviation of 86 minutes with a mean life of 505 minutes. Of the claim is true, in a sample of 120 batteries, what is the probability that the mean battery life would be greater than 523.8 minutes
Answers
Answer:
0.0082 = 0.82% probability that the mean battery life would be greater than 523.8 minutes
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Standard deviation of 86 minutes with a mean life of 505 minutes.
This means that [tex]\sigma = 86, \mu = 505[/tex]
Sample of 120:
This means that [tex]n = 120, s = \frac{86}{\sqrt{120}}[/tex]
What is the probability that the mean battery life would be greater than 523.8 minutes?
This is 1 subtracted by the p-value of Z when X = 523.8. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{523.8 - 505}{\frac{86}{\sqrt{120}}}[/tex]
[tex]Z = 2.39[/tex]
[tex]Z = 2.39[/tex] has a p-value of 0.9918.
1 - 0.9918 = 0.0082
0.0082 = 0.82% probability that the mean battery life would be greater than 523.8 minutes
which of the functions below could have created this graph?
A. F(x)=x^8+4x+20^2
B. F(x)=x^9+12x^8+x^7
C. F(x)=-x^6+x^3-x^2
D. F(x)=-x^7+4x^3-2x^2
Answers
Answer:
The answer is probably A, but you typed it in wrong so I do not know. I can only guess because the others do not seem to be correct either. I have graphed them on Desmos. The screenshot is below.
Step-by-step explanation: