Evaluate the arithmetic series:
40
Σ (6n - 9)
n=1
Answers
The arithmetic series has a value 11,880.
What dοes maths series class 10 entail?The difference at all between the twο successive integers in an arithmetic prοgressiοn (AP) sequence οf numbers is always the same amοunt. It is alsο knοwn as an arithmetic sequence.
An arithmetic sequence's nth term is determined by the fοrmula a = a + (n - 1)d. Thus, enter the numbers a = 2 & d = 3 intο tο the fοrmula tο determine the nth term. Step 2: In οrder tο determine the fifth term, change the nth term in the equatiοn tο be equal tο 5.
Tο evaluate the arithmetic series:
\(\rm 40 \Sigma (6n-9)n=1$\)
We first need tο find the sum οf the first n terms οf the sequence \(\rm (6n-9)\).
The fοrmula fοr the sum οf an arithmetic sequence is:
\(\rm S_n=n/2(a_1+a_n)$\)
where
Sₙ is the sum οf the first terms,
a₁ is the first term, and
aₙ is the nth term.
In this case, \(\rm a1=6(1)-9=-3$\), and \(\rm a n=6n-9.$\) Sο we can rewrite the sum as:
\(\rm 40 \Sigma (-3+6n)n=1$\)
Nοw we can use the fοrmula fοr the sum οf an arithmetic sequence tο find the sum οf the first n terms οf the sequence (-3 + 6n):
\(\begin{array}{l}{{\rm S n=n/2(a1+a n)}}\\\\ {{\rm S n=n/2(-3+6n)}}\\\\ {{\rm S n=-3n+3n^{2}/2}}\end{array}$\)
Sο the sum οf the first n terms οf the sequence.
Nοw we can substitute this expressiοn back intο the οriginal series:
\(\Rightarrow \rm 40 \Sigma (6n - 9) n=1 \\\\= \rm 40 ( \Sigma (-3 + 6n) n=1)\)
\(\rm = 40 (\Sigma -3 + \Sigma 6n) n=1\)
\(\rm = 40 (-3n + 6\Sigma n) n=1\)
\(\rm = 40 (-3n + 6(n(n+1)/2)) n=1\)
\(\rm = 40 (-3n + 3n^2 + 3n) n=1\)
\(\rm = 40 (3n^2) n=1\)
\(\rm = 120 (n^2) n=1\)
\(\rm 120 \Sigma n^2 n=1\)
Sο the series can be simplified tο \(\rm 120 \Sigma n^2 n=1\).
Using the fοrmula fοr the sum οf the squares οf the first n natural numbers:
\(\rm \Sigma n^2 =n(n+1)(2n+1)/6$\)
We can evaluate the series as:
\(\rm 120 \Sigma n^2 n=1 = 120 (\Sigma n^2 n=1)\)
\(\begin{array}{l}{\rm {=120(n(n+1)(2n+1)/6)n=1}}\\\\ {{=120(10(11)(21)/6)}}\\\\ {{=11,880}}\end{array}$\)
Therefοre, the value οf the arithmetic series is 11,880.
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Related Questions
Use >,<, or = to compare -2.7
-2.2.
Answers
Answer:
- 2.7 < - 2.2
Step-by-step explanation:
If a_1=3a 1 =3 and a_n=(a_{n-1})^2+4a n =(a n−1 ) 2 +4 then find the value of a_4a 4 .
Answers
The fourth term a₄ of the sequence is 29933
How to determine the value of a₄ of the sequence?From the question, we have the following definition of the function that can be used in our computation:
a₁= 3
aₙ = (aₙ₋₁)² + 4
The above definitions imply that we simply square the previous term and then add 4 to the squared result to get the current term
Using the above as a guide,
so, we have the following representation
a₂ = (3)² + 4
a₂ = 13
Also, we have
a₃ = (13)² + 4
a₃ = 173
Lastly, we have
a₄ = (173)² + 4
Evaluate the equation
a₄ = 29933
Hence, the value of a₄ is 29933
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Question 3 of 5
Given a sphere with a diameter of 6.2 cm, find its volume to the nearest whole
number.
OA. 39 cm³
OB. 125 cm³
OC. 70 cm³
OD. 998 cm³
Answers
Answer:
option B
125 cm³
Step-by-step explanation:
volume of sphere = 4/3πr³
where r = d/2
= 6.2/2 = 3.1
thus,
v = 4/3 * 3.14 * (3.1)³
= 4/3 * 3.14 * 29.79
= 124.72 cm³
rounding off to the nearest whole number,
= 125 cm³
The Cramer's Rule states that, in a system of equation in two unknowns, in which determinants D is not 0, X=_____and Y=_____.
Answers
Answer:
Step-by-step explanation:
\(x=\frac{D_{1}}{D} \\y=\frac{D_{2}}{D}\)
What is the Y intercept and write the slope intercept equation. Slope is 2.
Answers
Answer:
It looks like the Y intercept is 1 which is B
Step-by-step explanation:
Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.
5 cos4(x)
Answers
Using the power reducing formulas the expression \(5cos^4(x) = 5(1 + cos(2x))^{2/4}\).
What are power-reducing formula?We can define higher powers of a trigonometric function in terms of lower powers of the same function using the power-reducing formulas, which are trigonometric identities. Higher powers of cosine can be expressed using this formula in terms of sine and cosine's initial powers. For the other trigonometric functions, such as sine and tangent, there are equivalent formulas. These formulas can be used to solve trigonometric equations and to simplify trigonometric expressions.
The given function is 5cos^4(x).
Now, the power reducing formulas are given by:
\(cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1\\cos^2(x) = (1 + cos(2x))/2\)
Substituting the value of second equation in 1 we have:
cos(2x) = 2(1 + cos(2x))/4 - 1
\(cos(2x) = (2cos^2(x) - 1) = cos^2(x) - sin^2(x)\)
Now we have:
\(cos^2(x) = (1 + cos(2x))/2\)
Thus,
\(cos^4(x) = (1 + cos(2x))/2)^2\)
Now multiplying by 5:
\(5cos^4(x) = 5(1 + cos(2x))^{2/4}\)
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P=x-2 ÷ x+1 for whar value of x is P equal to zero
Answers
Answer:
x = 2
Step-by-step explanation:
P = \(\frac{x-2}{x+1}\)
P will equal zero when the numerator is equal to zero , that is
x - 2 = 0 ( add 2 to both sides )
x = 2
P = 0 when x = 2
The number of country A forces in country B decreased to approximately 34,000 in 2014 from a high of about 100,000 in 2008. The amount of country A funding for country B security forces also decreased during this period. The function f (x) = -1.330x + 5.149x +5.517 can be used to estimate the amount of country A funding for country B security forces, in billions of dollars, x years after January 2007. In what year was the amount of country A funding for country B security forces about $ 10.5 billion?
Answers
Answer:
2009
Step-by-step explanation:
The question can be interpreted as:
\(f(x) = -1.330x^2 + 5.149x + 5.517\)
Required
Determine x when f(x) = 10.5
Substitute 10.5 for f(x) in \(f(x) = -1.330x^2 + 5.149x + 5.517\)
\(10.5 = -1.330x^2 + 5.149x + 5.517\)
Collect Like Terms
\(-1.330x^2 + 5.149x + 5.517-10.5=0\)
\(-1.330x^2 + 5.149x -4.983=0\)
Using quadratic formula
\(x = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}\)
Where
\(a = -1.330\)
\(b = 5.149\)
\(c = -4.983\)
\(x = \frac{-5.149 \± \sqrt{5.149^2 - 4*-1.330*-4.983}}{2*-1.330}\)
\(x = \frac{-5.149 \± \sqrt{0.002641}}{-2.660}\)
\(x = \frac{-5.149 \± 0.0514}{-2.660}\)
Split:
\(x = \frac{-5.149 + 0.0514}{-2.660}\ or\ x = \frac{-5.149 - 0.0514}{-2.660}\)
\(x = \frac{-5.0976}{-2.660}\ or\ x = \frac{-5.2004}{-2.660}\)
\(x = 1.92\ or\ x = 1.96\)
Both values of x approximates to 2.
So:
\(x = 2\)
2 years after 2007 is 2009.
Hence, the funding is about $10.5 billion in 2009
Find the area of the triangle.
3 9
A
5
B
?] units²
![Find the area of the triangle.3 9A5B?] units](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/aWZMIZBnhz8tUKqZNrW2mkI5u8UBMICx.png)
Answers
The area of the triangle is 47.91 units²
How to find the area of the triangle?When all three sides of the triangle are known we can use Heron's formula. Consider the triangle ABC with sides a, b, and c has shown in the image.
Heron’s formula is:
Area =√s(s−a)(s−b)(s−c)
where,
a, b, c are the side length of the triangle
s is the semi-perimeter. s = (a+b+c)/2
In this case:
Using the knowledge of the radius of a circle. We can say:
a = BC = 3 + 9 = 12 units
b = AC = 5 + 3 = 8 units
c = AB = 5 + 9 = 14 units
s = (a+b+c)/2
s = (12+8+14)/2
s = 34/2
s = 17 units
Area = √s(s−a)(s−b)(s−c)
Area = √17(17−12)(17−8)(17−14)
Area = √17(5)(9)(3)
Area = √2295
Area = 47.91 units²
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the sum of two consecutive integers is less than 55. Fid the pair of integers with the greatest sum.
Answers
The pair of integers with the greatest sum where the sum of two consecutive integers is less than 55 is 26 and 27
How to determine the pair of integers with the greatest sum?The statement is given as:
the sum of two consecutive integers is less than 55.
Let the consecutive integers be x and y
Where y is greater than x
So, we have
y = x + 1
x + y < 55
Substitute y = x + 1 in the inequality x + y < 55
x + x + 1 < 55
Evaluate the sum
2x + 1 < 55
Subtract 1 from both sides
2x < 54
Divide both sides by 2
x < 27
The greatest integer less than 27 is 26.
So, we have
x = 26
Substitute x = 26 in y = x + 1
y = 26 + 1
Evaluate the sum
y = 27
So, we have
(26, 27)
Hence, the pair of integers with the greatest sum where the sum of two consecutive integers is less than 55 is 26 and 27
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In December 2016 the average price of unleaded
Answers
4.0/5.7=1.8/x what is the answer for u smart people
Answers
Answer:
here you go
Step-by-step explanation:
Answer:
2.565
Step-by-step explanation:
Hope that this helps!
if a person invested half of her money at 11% and half at 10% and received $420 interest. find the total amount
Answers
The total amount invested was $4000. $2000 was invested at 11% while the remaining $2000 was invested at 10%.
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
Let x represent the total amount of money invested, hence:
(0.5* x * 0.11) + (0.5 * x * 0.1) = 420
x = 4000
The total amount invested was $4000. $2000 was invested at 11% while the remaining $2000 was invested at 10%.
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Which phrase matches the algebraic expression below?
2(x + 7) + 10
A)Two times the sum of x and seven plus ten
B)Two times the difference of x and seven plus ten
C)Two times x minus seven plus ten
D)Two times x plus the sum of seven and ten
Answers
Answer:
the answer is a.
Step-by-step explanation:
two times the sum of x and seven plus ten
2(x+7)+10
Frances has 4 square baking dishes. Each pan is 7 inches × 7 inches × 2 inches. Which expression will give her the total volume of all the pans? A. 7^2 × 2 B. 7 × 2^4 C. (7 × 2 × 2) × 4 D. (7^2 × 2) × 4
Answers
Answer:
D. (7^2 X2) x 4
The equation inside the parenthesis finds the volume of one pan and then that is multiplied by 4 which is the total number of pans. This will find the volume of all 4 pans.
Answer: (7^2 × 2) × 4
Step-by-step explanation:
In the diagram below, FC = 10.9,
DE = 17.5, and DF = 13.1. Find the
length of EB. Round your answer to the
nearest tenth if necessary.
D
F
E
C
B
Answers
The length of EB is approximately 14.6 units when rounded to the nearest tenth.
To find the length of EB, we can use the property of similar triangles in this diagram. By looking at triangle DFE and triangle CFB, we can see that they are similar triangles.
Using the similarity ratio, we can set up the proportion:
DF / CF = DE / EB
Plugging in the given values, we have:
13.1 / 10.9 = 17.5 / EB
To find EB, we can cross-multiply and solve for EB:
13.1 * EB = 10.9 * 17.5
EB = (10.9 * 17.5) / 13.1
EB ≈ 14.6
Therefore, the length of EB is approximately 14.6 units when rounded to the nearest tenth.
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A menu has five choices of appetizers
Answers
The number of the meals possible is 350.
The complete question is given below:-
A menu has 5 choices of appetizers, ten main courses, and seven desserts. How many meals are possible?
What is the combination?The arrangement of the different things or numbers in a number of ways is called the combination.
Given that:-
A menu has 5 choices of appetizers, ten main courses, and seven desserts.The number of the combination of the meals will be calculated as:-
N = 5 x 10 x 7
N = 350
Therefore the number of the meals possible is 350.
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what is the length of the hypotenuse of a right triangle whose legs have lengths 12 cm and 16 cm?
Answers
In Pythagorean Theorem, the hypotenuse of a right triangle is always the longest side and is representative of a variable we know as c^2, or c squared. Since we know that the right triangle in this problem has lengths 12 cm and 16 cm, we know that a^2 equals 12 squared and b^2 equals 16 squared. Because the formula for Pythagorean Theorem is a^2 + b^2 = c^2, we can write our problem out:
12^2 + 16^2 = c^2
12^2 (or 12 squared) equals 144 since 12 x 12 = 144
16^2 (or 16 squared) equals 256 since 16 x 16 = 256
Since we know what a squared and b squared equal, we can add them to find what our hypotenuse squared is.
144 + 256 = 400
We now know that our hypotenuse squared equals 400. To find c, we can simply find the square root of 400. You can keep this question in mind when doing it: “What number can be multiplied by itself to get 400?”
When calculating, we will find that the square root of 400 is 20. To verify, we can multiply 20 by itself to see what we get.
20 x 20 = 400
Our equation is true! Therefore, here is your final equation:
12^2 + 16^2 = 20^2
The length of the hypotenuse is 20 cm. If you need any extra help, let me know and I will gladly assist you.
Which number is the same as 125,000,000
a) 1.25 x 10 to the power of 6 c) .125 x 10 to the power of 6 e) 125 million
b) 125,000 million d) 1.25 million
Answers
Answer:
c
Step-by-step explanation:
I'm pretty sure the answer is c
Write a rule for the sequence. 1/9, 1/3, 5/9 Pls hurry Pls S's S's Sissies S's See fast
Answers
Two trains, Train A and Train B, weigh a total of 389 tons. Train A is heavier than Train B. The difference of their weights is 263 tons. What is the weight of each train? Equation 1: ____________ Equation 2: ___________ Train A weight: ________ Train B weight: ________
Answers
(
4
x
−
3
)
(
x
2
+
2
x
−
6
)
Answers
Answer:
120
Step-by-step explanation:
3x+(8x-16) simplest form
Answers
Step-by-step explanation:
3x × 8x- 3x×16
24x -48x
-24x
A sample has a mean of M = 90 and a standard devia-tion of s 20.=a. Find the z-score for each of the following X values.X = 95X = 80X = 98X = 88X = 105X = 76
Answers
Answer:
\(\begin{gathered} X=95,z=0.25 \\ X=80,z=-0.5 \\ X=98,z=0.4 \\ X=88,z=-0.1 \\ X=105,z=0.75 \\ X=76,z=-0.7 \end{gathered}\)Explanation:
Given a sample with the following:
• Mean,M = 90
,• Standard deviation, s = 20
To find the z-score for each of the given X values, we use the formula below:
\(\begin{equation*} z-score=\frac{X-\mu}{\sigma}\text{ where }\begin{cases}{X=Raw\;Score} \\ {\mu=mean} \\ {\sigma=Standard\;Deviation}\end{cases} \end{equation*}\)The z-scores are calculated below:
\(\begin{gathered} \text{When X=95, }z=\frac{95-90}{20}=\frac{5}{20}=0.25 \\ \text{When X=80, }z=\frac{80-90}{20}=\frac{-10}{20}=-0.5 \\ \text{When X=98, }z=\frac{98-90}{20}=\frac{8}{20}=0.4 \end{gathered}\)\(\begin{gathered} \text{When X=88,}z=\frac{88-90}{20}=\frac{-2}{20}=-0.1 \\ \text{When X=105, }z=\frac{105-90}{20}=\frac{15}{20}=0.75 \\ \text{When X=76, }z=\frac{76-90}{20}=\frac{-14}{20}=-0.7 \end{gathered}\)Nate has 6 3/5 of fabric. He uses 3 1/2 yards of fabric to make a pillow. How much fabric dose he left
Answers
Answer:
3 and 1/2 minus 2/3 minus 1/3 =
7/2 minus 2/3 minus 1/3 =
21/6 minus 4/6 minus 2/6 =
17/6 - 2/6 =
15/6 = 5/2 = 2 and 1/2
Optionally, you can think of it this way:
He used 2/3 yards for the pillow and 1/3 yard for the scarf.
So he used 1 yard
3 and 1/2 minus 1 = 2 and 1/2 = 5/2
Step-by-step explanation:
We know that Nate has 6 3/5 to begin with. He used 3 1/2. So, 6 3/5 - 3 1/2 = 3 1/10.
here's the question ...........
Answers
Answer:
x intercept -7 and y intercept is 9
Can the following quadrilateral be proven to be a parallelogram based on the given information? Explain.
(picture shown below)
A.) No. It cannot be proven because at least two of the adjacent angles are not congruent to each other
B.) Yes. It can be proven because both pairs of opposite sides are congruent.
C.) No. It cannot be proven because it does not have an angle that is supplementary to both of its consecutive angles.
D.) Yes. It can be proven because both pairs of opposite angles are congruent.
Answers
Answer:
Step-by-step explanation:
The answer is, no!
Step-by-step explanation:
The reason to that answer is that some other shapes are also, having the same properties as shown. An example for that property would also mean a rectangle and not always a parallelogram.
So, here is the given information on what we need to classify a shape as a parallelogram:-
1) Parallel Sides
2)Opposite Sides are equal.
3) Adjacent sides are not equal.
4) Opposite angles are equal.
5)Adjacent angles are not equal.
Based on this, we can classify a shape as a parallelogram.
Hope, this answer helps!
URGENT! Can someone please help?
Answers
a. The missing values of the logarithm expression is log₃(40).
b. The missing values of the logarithm expression is log₅(8).
c. The missing values of the logarithm expression is log₂(1/25).
What is the missing of the logarithm expression?The missing values of the logarithm expression is calculated as follows;
(a). log₃5 + log₃8, the expression is simplified as follows;
log₃5 + log₃8 = log₃(5 x 8) = log₃(40)
(b). The log expression is simplified as;
log₅3 - log₅X = log₅3/8
log₅X = log₅8
X = 8
(c). The log expression is simplified as;
-2log₂5 = log₂Y
log₂5⁻² = log₂Y
5⁻² = Y
1/25 = Y
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if x-y = 8 and xy=5 , find x^2 + y^2
Answers
Answer:
x² + y² = 74
Step-by-step explanation:
given
(x - y) = 8 ( square both sides )
(x - y)² = 8² ← expand left side using FOIL
x² - 2xy + y² = 64 ← substitute xy = 5
x² - 2(5) + y² = 64
x² - 10 + y² = 64 ( add 10 to both sides )
x² + y² = 74
income from operations of $231,000. In addition, it received interest income of $23,100 and received dividend income of $29,700 from another corporation. Finally, it paid $11,000 of interest income to its bondholders and paid $45,000 of dividends to its common stockholders. The firm's federal tax rate is 21%. What is the firm's federal income tax? Do not round intermediate calculations. Round your answer to the nearest dollar.
Answers
The firm's federal income tax is $57,168.
To calculate the firm's federal income tax, we need to determine its taxable income first. Taxable income is calculated by subtracting deductions from the firm's total income.
In this case, the firm's total income consists of income from operations, interest income, and dividend income.
Total income:
Income from operations = $231,000
Interest income = $23,100
Dividend income = $29,700
Total income = $231,000 + $23,100 + $29,700 = $283,800
Next, we deduct expenses from the total income. In this case, the only relevant expense is the interest paid to bondholders:
Total expenses = Interest paid to bondholders = $11,000
Taxable income = Total income - Total expenses = $283,800 - $11,000 = $272,800
Now, we can calculate the federal income tax by applying the federal tax rate of 21% to the taxable income:
Federal income tax = Taxable income * Federal tax rate
Federal income tax = $272,800 * 0.21 = $57,168
Therefore, the firm's federal income tax is $57,168.
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