Let a be the first term and d the common difference between consecutive terms. Then the next few terms in the sequence are
a + d
a + 2d
a + 3d
and so on, up to the k-th term
a + (k - 1) d
The sum of the first n terms of this sequence is
[tex]\displaystyle S_n = \sum_{k=1}^n(a+(k-1)d) = 12n-2n^2[/tex]
Expanding the sum, we have
[tex]\displaystyle S_n = \sum_{k=1}^n (a+(k-1)d) \\\\ S_n = \sum_{k=1}^n(a-d+dk) \\\\ S_n = (a-d)\sum_{k=1}^n1+d\sum_{k=1}^nk \\\\ S_n = (a-d)n+\frac{d}2n(n+1) \\\\ S_n = (a-d)n+\frac{d}2(n^2+n) \\\\ S_n = \left(a-\frac{d}2\right)n+\frac{d}2n^2[/tex]
It follows that
a - d/2 = 12
d/2 = -2
Solve these equations for a and d.
d/2 = -2 ==> d = -4
a - d/2 = a + 2 = 12 ==> a = 10
So the sequence is
10, 6, 2, -2, -6, -10, -14, …
k = 2; f(x) = 2x3 + 3x2 - 4x + 4; Lower bound? (1 point)
The sum of the reciprocals of two consecutive even integers is 3/4
Find the two integers.
[tex] \Large{ \underline{ \underline{ \bf{ \orange{Solution:}}}}}[/tex]
Let one of those even numbers be x, Then other even number would be x + 2.
According to question,
⇛ Their reciprocal add upto 3/4
So, we can write it as,
⇛ 1/x + 1/x + 2 = 3/4
⇛ x + 2 + x / x(x + 2) = 3/4
⇛ 2x + 2 / x² + 2x = 3/4
Cross multiplying,
⇛ 3(x² + 2x) = 4(2x + 2)
⇛ 3x² + 6x = 8x + 8
⇛ 3x² - 2x - 8 = 0
⇛ 3x² - 6x + 4x - 8 = 0
⇛ 3x(x - 2) + 4(x - 2) = 0
⇛ (3x + 4)(x - 2) = 0
Then, x = -4/3 or 2
☃️ It can't be -4/3 because it is fraction and negative number. So, x = 2
Then, x + 2 = 4
✤ So, The even numbers are 2 and 4.
━━━━━━━━━━━━━━━━━━━━
Which of the binomials below is a factor of this expression?
16x2 + 40xy + 25y2
16x²+40xy + 25y²
(4x+5y) (4x+5y)
(4x+5y)²
I hope I helped you^_^
Two numbers are in the ratio 2:3. If 3 is added to the numbers, the ratio changes to 3:4. Find the numbers.
Answer:
6:9
Step-by-step explanation:
Answer:
6 and 9
Step-by-step explanation:
according to the question, the nos. are 2x and 3x.
then, (2x+3)/(3x+3)=3/4
therefore, 4(2x+3)=3(3x+3)
8x+12=9x+9
12-9=9x-8x
3=x
therefore the nos. are 2×3=6 & 3×3=9
Answer from Gauth math
Write expression for the sum x and 6
Answer:
X+6
Step-by-step explanation:
Sum means Addition.
. En el triángulo ABC, la medida del ángulo exterior en el vértice B es el triple de la medida del ángulo C y la mediatriz de BC corta a AC en el punto F. sabiendo que FC=12. Calcular AB.
Responder:
| AB | = 12m
Explicación paso a paso:
Verifique el diagrama en el archivo adjunto.
En el diagrama, se puede ver que el lado FC es igual al lado FB de acuerdo con el triángulo isósceles FBC.
Además, el lado FB es igual a AB ya que son paralelos entre sí.
De la declaración anterior, | FC | = | FB | y | FB | = | AB |
Esto significa | FC | = | FB | = | AB |
Por lo tanto desde | FC | = 12 m, | AB | = 12 m ya que ambos lados son iguales.
De ahí el lado | AB | se mide 12m
Use the information angle 8 is congruent to angle 11 to determine which lines are parallel.
A. p || q
B. l || m
C. m || n
D. l || n
Answer:
A
Step-by-step explanation:
based on line p and q
Answer: p || q
Or A
Step-by-step explanation:
good luck
Help me with this question plz
19. 68 because 90seconds 1hr 30 mons
Statistics professors believe the average number of headaches per semester for all students is more than 18. From a random sample of 15 students, the professors find the mean number of headaches is 19 and the standard deviation is 1.7. Assume the population distribution of number of headaches is normal.
Complete Question
Statistics professors believe the average number of headaches per semester for all students is more than 18. From a random sample of 15 students, the professors find the mean number of headaches is 19 and the standard deviation is 1.7. Assume the population distribution of number of headaches is normal.the correct conclusion at [tex]\alpha =0.001[/tex] is?
Answer:
There is no sufficient evidence to support the professor believe
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 18[/tex]
The sample size is [tex]n = 15[/tex]
The sample mean is [tex]\= x = 19[/tex]
The standard deviation is [tex]\sigma = 1.7[/tex]
The level of significance is [tex]\alpha = 0.001[/tex]
The null hypothesis is [tex]H_o: \mu = 18[/tex]
The alternative hypothesis is [tex]H_a : \mu > 18[/tex]
The critical value of the level of significance from the normal distribution table is
[tex]Z_{\alpha } = 3.290527[/tex]
The test hypothesis is mathematically represented as
[tex]t = \frac{\= x - \mu }{ \frac{\sigma}{ \sqrt{n} } }[/tex]
substituting values
[tex]t = \frac{ 19 - 18}{ \frac{1.7}{ \sqrt{15} } }[/tex]
[tex]t = 2.28[/tex]
Looking at the value of t and [tex]Z_{\alpha }[/tex] we can see that [tex]t < Z_{\alpha }[/tex] so we fail to reject the null hypothesis.
This mean that there is no sufficient evidence to support the professor believe
Please Help
Function 1 is defined by the equation: p=r+7
Function 2 is defined by the table shown in the image below
Which function has a greater slope, function 1 or function 2?
Answer:
The slope of Function 2 (m=1.1) is greater than the slope of Function 1 (m=1).
Step-by-step explanation:
First, note that p is essentially the y and that r is the x. Thus, to make this easier to see, convert p to y and r to x. Thus:
[tex]y=x+7[/tex]
From the above equation, we can determine that the slope is 1. Thus, the slope of Function 1 is 1.
To find the slope of the table, simply use the slope formula. Use any two points. I'm going to use the points (0,8) and (10,19). Let (0,8) be x₁ and y₁, and (10,19) be x₂ and y₂. Therefore:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{19-8}{10-0}=11/10=1.1[/tex]
Thus, the slope of Function 2 is 1.1.
1.1 is greater than 1.
Thus, the slope of Function 2 is greater than the slope of Function 1.
Answer:
Function 2 has the greater slope
Step-by-step explanation:
Factor 13ab3 + 39a2b5.
[tex]13ab^3+39a^2b^5\\\\\boxed{\boxed{\boxed{13ab^3(1+3ab^2)}}}\\\\[/tex]
Brazil number one.
Answer:
there's no answer for that equation
Evaluate the expression: -(31 + 2) +7² - (-5²)
A) -9
B) -5
C) 41
OD -40
Answer: C. 41
Step-by-step explanation:
[tex]-\left(31+2\right)+7^2-\left(-5^2\right)[/tex]
[tex]=-33+7^2-\left(-5^2\right)[/tex]
[tex]\left(-5^2\right)=-25[/tex]
[tex]=-33+7^2-\left(-25\right)[/tex]
[tex]7^2=49[/tex]
[tex]=-33+49-\left(-25\right)[/tex]
[tex]-33+49=16[/tex]
[tex]=16-\left(-25\right)[/tex]
[tex]\mathrm{Apply\:rule\:}-\left(-a\right)\:=\:+a[/tex]
[tex]16+25=41[/tex]
A family has two cars. The first car has a fuel efficiency of 35 miles per gallon of gas and the second has a fuel efficiency of 15 miles per gallon of gas. During one particular week, the two cars went a combined total of 1825 miles, for a total gas consumption of 75 gallons. How many gallons were consumed by each of the two cars that week?
Answer:
1) 35 gallons by the first car 2) 40 gallons by the second car
Step-by-step explanation:
Suppose the first car used x gallons, when the second car used the rest- 75-x
If the first car's efficiency is 35 miles per galon, its milleage is 35*x, the second car's milleage is 15*(75-x). And the summary milleage is equal to 1825.
35x+15(75-x)=1825
35x+1125-15x= 1825
20x=700
x=35- gallons consumed by the first car,
75-35=40- gallons consumed by the second one
Let A and B be any two sets. Show that:
Show that (
AUB)', (BUA)' = 0
Step-by-step explanation:
(AUB)' means they are all outside the set A and B so thats 0. Hope it helps
Fill in the blanks
To factor the polynomial 3x2–5x - 12, find two numbers whose product is
and
whose sum is
Answer:
Step-by-step explanation:
Ooooooofkfnvkanggkmfifkfkfkdkcknavnhkgnvkic
Work out the values of a, b and k ? 30 points
Answer:
[tex]\displaystyle a=4, b= \frac{25}{4}, \text{ and } k = \frac{125}{2}[/tex]
Step-by-step explanation:
Note that the graph passes through the points: (0, 4), (1, 25), and (1.5, k).
The standard exponential function has the form:
[tex]\displaystyle y = ab^x[/tex]
The point (0, 4) tells us that y = 4 when x = 0. Therefore:
[tex](4) = a(b)^0[/tex]
Since anything raised to zero is one:
[tex]a=4[/tex]
Hence, our function is now:
[tex]y = 4(b)^x[/tex]
The point (1, 25) tells us that y = 25 when x = 1. By substituting:
[tex](25) = 4(b)^{(1)}[/tex]
Solve for b:
[tex]\displaystyle b = \frac{25}{4}[/tex]
Thus, our completed function is:
[tex]\displaystyle y = 4\left(\frac{25}{4}\right)^x[/tex]
To find k, simply substitute 1.5 for x. This yields:
[tex]\displaystyle y = k = 4\left(\frac{25}{4}\right)^{(1.5)}[/tex]
And evaluate. Hence:
[tex]\displaystyle \begin{aligned} k &= 4\left(\frac{25}{4}\right)^{3/2} \\ \\ &= 4\left(\left(\frac{25}{4}\right)^{1/2}\right)^3 \\ \\ &= 4\left(\frac{5}{2}\right)^3 \\ \\ &= 4\left(\frac{125}{8}\right) \\ \\ &= \frac{125}{2}\end{aligned}[/tex]
In conclusion:
[tex]\displaystyle a=4, b= \frac{25}{4}, \text{ and } k = \frac{125}{2}[/tex]
PRACTICE ANOTHER A piece of wire 18 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? m (b) How much wire should be used for the square in order to minimize the total area? m
Answer:
Step-by-step explanation:
We have the equations
4x + 3y = 18 where x = the side of the square and y = the side of the triangle
For the areas:
A = x^2 + √3y/2* y/2
A = x^2 + √3y^2/4
From the first equation x = (18 - 3y)/4
So substituting in the area equation:
A = [ (18 - 3y)/4]^2 + √3y^2/4
A = (18 - 3y)^2 / 16 + √3y^2/4
Now for maximum / minimum area the derivative = 0 so we have
A' = 1/16 * 2(18 - 3y) * -3 + 1/4 * 2√3 y = 0
-3/8 (18 - 3y) + √3 y /2 = 0
-27/4 + 9y/8 + √3y /2 = 0
-54 + 9y + 4√3y = 0
y = 54 / 15.93
= 3.39 metres
So x = (18-3(3.39) / 4 = 1.96.
This is a minimum value for x.
So the total length of wire the square for minimum total area is 4 * 1.96
= 7.84 m
There is no maximum area as the equation for the total area is a quadratic with a positive leading coefficient.
The length of the square must be 4 m in order to maximize the total area.
What are the maxima and minima of a function?When we put the differentiation of the given function as zero and find the value of the variable we get maxima and minima.
We have,
Length of the wire = 18 m
Let the length of the wire bent into a square = x.
The length of the wire bent into an equilateral triangle = (18 - x)
Now,
The perimeter of a square = 4 side
4 side = x
side = x/4
The perimeter of an equilateral triangle = 3 side
11 - x = 3 side
side = (18 - x)/3
Area of square = side²
Area of equilateral triangle = (√3/4) side²
Total area:
T = (x/4)² + √3/4 {(18 -x)/3}² _____(1)
Now,
To find the maximum we will differentiate (1)
dT/dx = 0
2x/4 + (√3/4) x 2(18 - x)/3 x -1 = 0
2x / 4 - (√3/4) x 2(18 - x)/3 = 0
2x/4 - (√3/6)(18 - x) = 0
2x / 4 = (√3/6)(18 - x)
√3x = 18 - x
√3x + x = 18
x (√3 + 1) = 18
x = 18 / (1.732 + 1)
x = 18/2.732
x = 6.58
x = 7
Thus,
The length of the square must be 7 m in order to maximize the total area.
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Need help please will mark brainliest
Step-by-step explanation:
Maximum = 62
Median = (34+37+39+32+48+45+53+62+58+61+60+41)/12= 47.5≈48
quartile
In increasing order
32, 34, 37, 39, 41, 45, 48, 53, 58, 60, 61, 62
Upper quartile= (58+60)/2 = 59
Lower quartile= (37+39)/2 = 38
Minimum= 32
Find the missing side of the triangle.
Answer:
x = 2(sqrt26) or x = 10.1980390272...
Step-by-step explanation:
Right triangles can be solved using Pythagorean theorem, where the legs are squared and added together to get the hypotenuse's length squared.
If we set up the equation:
11^2 + x^2 = 15^2
which is
121 + x^2 = 225
subtract 121 from both sides
x^2 = 104
sqrt both sides
x = 2(sqrt26), or x = 10.1980390272...
Answer:
Using Pythagoras theorem
c²=a²+b²
(15)²=x² +(11)²
225=x²+ 121
-x²=121-225
-x²=-104
multiply both sides by (-1)
x²=104
x=✓104
x=10ft
please help me
no links or files
thank you !
Jane is saving to buy a cell phone. She is given a $100 gift to start and saves $35 a month from her allowance. So after one month, Jane has saved $135. Does it make sense to represent the relationship between the amount saved and the number of months with one constant rate? Why or why not? Explain your answer.
Jane is given a $100 gift to start and saves $35 a month from her allowance.
After 1 month, Jane has saved
After 2 months, Jane has saved
After three months, Jane has saved
and so on
In general, after x months Jane has saved
This means that it makes sense to represent the relationship between the amount saved and the number of months with one constant rate (in this case the constant rate is 35). It makes sense because the amount of money increases by $35 each month. Since the amount of increase is constant, we get constant rate. Also the initial amount is known ($100), so there is a possibility to write the equation of linear function representing this situation.
Step-by-step explanation:
If x+1/x = 12, find the value of x^2+1/x^2
[tex]\boxed{\sf (a+b)^2=a^2+b^2+2ab}[/tex]
Now
[tex]\\ \sf\longmapsto x+\dfrac{1}{x}=12[/tex]
[tex]\\ \sf\longmapsto \left(x+\dfrac{1}{x}\right)^2=12^2[/tex]
[tex]\\ \sf\longmapsto x^2+2\times x\times \dfrac{1}{x}+\left(\dfrac{1}{x}\right)^2=144[/tex]
[tex]\\ \sf\longmapsto x^2+\dfrac{1}{x^2}+2=144[/tex]
[tex]\\ \sf\longmapsto x^2+\dfrac{1}{x^2}=144-2=142[/tex]
Answer:
122
Step-by-step explanation:
x+1/x = 12
x + 1 = 12x
x = 1/11
（1/11）² + 1 / （1/11）² = 1/121 +1 /1/121 = 122
Jack is 4 times as old as Lacy. 3 years from now the sum of their ages will be 71 . How old are they now?
Answer:
Lacy is 13 and Jack is 52
Step-by-step explanation:
In 3 years their ages will add up to 71 so you have to subtract 6 as there are two of them to get 65. Lacy's age is represented by x and since Jack is 4 times older his age is represented by 4x. So added together their age is 5x. So 5x=65. Then 65/5=13. So 13=x. So Lacy is 13 and Jack is 52 as 13x4 is 52.
Consider exponential function h.
h(x) = 3x + 4
The function is always positive.
(0,5) is the y-intercept, since the graphed line never crosses the x axis, there is no x-intercept.
The function is positive and greater than 4 for all values of x
Not sure what the actual choices are on a couple of the questions. The choices would help answering.
In a study of academic procrastination, researchers reported that for a random sample of 41 undergraduate students preparing for a psychology exam, the mean time spent studying was 11.9 hours with a standard deviation of 4.5 hours. Compute a 95% confidence interval for μ, the mean time spent studying for the exam among all students taking this course.
Answer:
The 95% confidence interval is [tex]10.5 < \mu <13.3[/tex]
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 41[/tex]
The sample mean is [tex]\= x = 11.9 \ hr[/tex]
The standard deviation is [tex]\sigma = 4.5[/tex]
For a 95% confidence interval the confidence level is 95%
Given that the confidence level is 95% then the level of significance can be mathematically represented as
[tex]\alpha = 100 - 95[/tex]
[tex]\alpha = 5 \%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical values of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table
The values is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{ \sqrt{n} }[/tex]
substituting values
[tex]E = 1.96 * \frac{ 4.5 }{ \sqrt{41} }[/tex]
[tex]E = 1.377[/tex]
The 95% confidence interval is mathematically represented as
[tex]\= x - E < \mu < \= x - E[/tex]
substituting values
[tex]11.9 - 1.377 < \mu <11.9 + 1.377[/tex]
[tex]10.5 < \mu <13.3[/tex]
Reduce to simplest form.
-3/2 - 3/8
Answer:
hope this help you a lot
have a great day
f x equals 1 / x - 3 + 7 find the inverse of f x and its domain
Answer:
A
Step-by-step explanation:
f(x) = 1/(x-3)+7
f(x)-7=1/(x-3)
x=1/(f(x)-7)+3
f^-1(x)=1/(x-7)+3, where x≠7
The correct answer is option (a) [tex]f^{-1}(x)= \frac{1}{x-7} +3[/tex] where [tex]x\neq 7[/tex].
DomainThe domain of a function is the complete set of possible values of the independent variable
How to find domain?Given [tex]f^{}(x)= \frac{1}{x-3} +7[/tex]
Let
[tex]y= \frac{1}{x-3} +7[/tex]
⇒y-7= 1/x-3
⇒x-3 =1/y- 7
⇒ [tex]x= \frac{1}{y-7} +3[/tex]
hence option a is correct
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Please answer this correctly without making mistakes
Answer:
1/4 miles
Step-by-step explanation:
Hey there!
Well starting at Campbell and going to Morristown it is 1/4 miles.
Going from Campbell to Clarksville it is 2/4 miles.
So to find the difference we’ll subtract.
2/4 - 1/4
= 1/4 miles
Hope this helps :)
Biểu diễn 1 ràng buộc trên mặt phẳng tọa độ Oxy là
Step-by-step explanation:
yah kaun se language mein likha hai aapane
Find X using the Angle Sum Theorem
Answer:
x = 20°
Step-by-step explanation:
So when I learned it we called it the exterior angle theorem not the angle sum theorem but here goes.
Since exterior angle = 110 Degrees,
--> The Inner 2 angles's sum = 110 Degrees
so, 70 + 2x = 110
=> 2x = 40
x = 20
x = 20°
Hope this helps!
!2,19,26 what comes nxt
Answer:
12 , 19 , 26 , 33
Explaination:Here, n+7
12+7=19
19+7=26
So,
26+7=33
Hope you understand ❣
Step-by-step explanation:
12 , 19 , 26 , ?
Given
___________
a1= 12
a2= 19
a3 = 26
d= ?
a4 = ?
––——————
we can solve this by using formula from Ap .
But for this we have to find d
As we know that
common difference(d) = a2-a1 = 19 -12
= 7
so difference after every no is 7 so
a4 = a3 + d
= 26 +7
= 33
So 33 is ur answer mate
Hope it helps