The summation expression in the following series has an absolute value in it. Expand and evaluate the summation notation. What is the sum of the series?
Answers
We have that [tex]3<\dfrac{10}3<4[/tex], which means [tex]|3i-10|[/tex] reduces to [tex]3i-10[/tex] when [tex]i\ge4[/tex], or reduces to [tex]10-3i[/tex] when [tex]i\le3[/tex].
So we can expand the summation as
[tex]\displaystyle\sum_{i=1}^6|3i-10|=\sum_{i=1}^3(10-3i)+\sum_{i=4}^6(3i-10)[/tex]
Notice that the 10 contributes a total of 30 from the first sum, and -30 from the second sum, so those terms cancel, leaving us with
[tex]\displaystyle3\left(\sum_{i=4}^6i-\sum_{i=1}^3i\right)=3((4+5+6)-(1+2+3))=3(15-6)=3(9)=27[/tex]
To expand and evaluate the summation notation with an absolute value, we can use the binomial theorem. The expanded form of the series will depend on whether the sum inside the absolute value is positive or negative. Substituting the given values and simplifying the expression will give the sum of the series.
Explanation:A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In this case, we have an absolute value in the summation expression.
To expand and evaluate the summation notation, we can use the binomial theorem.
The binomial theorem states that (a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n, where C(n,k) represents the binomial coefficient.
In the given case, we have an expression of the form |a + b|.
To expand this, we can consider two cases: a + b ≥ 0 and a + b < 0. If a + b ≥ 0, then |a + b| = a + b. If a + b < 0, then |a + b| = -(a + b).
Thus, the expanded form of the series will be a + b when a + b ≥ 0, and -(a + b) when a + b < 0.
To evaluate the series, substitute the given values for a and b into the expanded form and simplify the expression.
Related Questions
What would be the radius of A= 530.66 square meter
Answers
√169=√r²
13=r
Help!! There are at least 3 slices of bread left in the bag how do I write an inequality for that?
Answers
x = slices of bread
x ≥ 3
the "≥" means greater than or equal, which basically means at least
Answer:
I need help on this question too but instead of 3 its 7!!! i also need to write both a inequality and a graph! pls respond as fast as possible.
Step-by-step explanation:
thanks
The local sports shop has backpacks on sale for $28. The original price was $40. What is the percent decrease for the backpack price?
Answers
Which of the following is not a way to represent the solution of the inequality 2(x − 2) less than or equal to 2?
A number line with a closed circle on 3 and shading to the right
A number line with a closed circle on 3 and shading to the left
3 greater than or equal to x
x less than or equal to 3
Answers
Answer:
The correct option is 1.
Step-by-step explanation:
The given inequality is
[tex]2(x-2)\leq 2[/tex]
Divide both the sides by 2.
[tex]\frac{2(x-2)}{2}\leq \frac{2}{2}[/tex]
[tex](x-2)\leq 1[/tex]
Add 2 on both the sides.
[tex]x-2+2\leq 1+2[/tex]
[tex]x\leq 3[/tex]
The value of x is than or equal to 3. In other words 3 greater than or equal to x.
Since the sign of inequality is ≤, therefore a number line with a closed circle on 3 and shading to the left represent the above inequality.
Only the statement "A number line with a closed circle on 3 and shading to the right" is not is not a way to represent the solution of the given inequality.
Therefore the correct option is 1.
Quadrilateral ABCD is inscribed in a circle.
What is the measure of angle A?
Enter your answer in the box.
m∠A= ____°
Answers
8x+10+2x+30=360
10x=320
x=32
∠A=4×32+5=133º
Answer:
133
Step-by-step explanation:
Given is a cyclic quadrilateral ABCD.
We know that in a cyclic quadrilateral, opposite angles are supplementary.
A+C =180
Given that A = 4x+5 and C = x+15
Add these and equate to 180
[tex]4x+5+x+15 =1805x=180-20 =160x=32A= 4(32)+5 =133[/tex]
Verify:
When x= 32, Angle C = 47
A+C = 133+47 = 180
Hence right answer
I need help I would really thank wheeler answers this for me
Answers
a) 36 = 6*6 = 2*3*2*3 = 2^2*3^2
b) 45 = 5*9 = 5*3*3 = 3^2*5
c) 2 is not a common factor; 3^2 is a common factor; 5 is not a common factor. The greatest common factor is 3^2 = 9.
d) The least common multiple can be found a couple of ways.
.. 1) the product of the distinct factors to their highest powers
.. .. .. 2^2 * 3^2 * 5 = 180
.. 2) the product of the numbers divided by their greatest common factor.
.. .. .. 36 * 45 / 9 = 180
Find the sum of 5 smallest consecutive whole numbers.
Answers
0+1+2+3+4= 10
that is the answer
Answer:
10
Step-by-step explanation:
these are so hard please help me and answer the last 4 I posted
Answers
6 days = 1 shed
1 day = 1/6 shed
Mark:
8 days = 1 shed
1 day = 1/8 shed
Together:
1 day = 1/6 + 1/8 shed
Ans: 1/6d + 1/8d = 1
Kaitlyn = 1/6
Mark = 1/8
Then,
d = number of days.
The equation then if they work together is:
1 / 6d + 1 / 8d = 1
answer:
the equation for number of days it would take to build the sheld together is
1 / 6d + 1 / 8d = 1
beth owes 17,000 on her car loan. The interest rate on her loan is 4.25% She will be paying this loan off for 4 years. How much will she pay altogether?
Answers
To find the total payment Beth will make on her car loan, calculate the total cost using the loan formula. Beth will pay a total of $19,890 in total.
Explanation:Beth owes $17,000 on her car loan with a 4.25% interest rate, to be paid off over 4 years. To calculate the total amount she will pay, you use the formula for calculating the total cost of a loan: Total Payment = Principal + (Principal x Interest Rate x Time).
Principal = $17,000Interest Rate = 4.25% = 0.0425Time = 4 years.Plugging in the values:
Total Payment = $17,000 + ($17,000 x 0.0425 x 4)
Beth will pay a total of $17,000 + ($17,000 x 0.0425 x 4) = $17,000 + $2890 = $19,890 altogether.
Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)
Answers
1. Probability=1-[probability_all_boys].
for [probability_all_boys]=1/[all_possible_combinations];
for [all_combinations]=2³=8. It means that
Probability=1-(1/8)=7/8=0.875.
2. Probability=[needed_cases]\[all_possible_cases];
all possible combinations are:
bbb;ggg;bgg;bgb;bbg;gbg;gbb;ggb; - (where 'g' - girl, 'b' - boy) total 8.
needed cases: ggg;bgg;bgb;bbg;gbg;gbb;ggb, - total 7.
Probability=7/8=0.875
Final answer:
The probability of having at least one girl when a couple has three children can be found by calculating the probability of having all boys and subtracting it from 1.
Explanation:
Probability that at least one child is a girl:
Find the probability of having all boys: (1/2)³ = 1/8
Subtract this from 1 to get the probability of having at least one girl: 1 - 1/8 = 7/8
Isabella jump rope is 2.03 meters long. Avery jump rope is 0.15 meters shorter than Idabellas jump rope. Measures in meters, what is the length of Averys jump rope?
Answers
Isabella's jump rope is 2.03 metres long, and Avery's is 0.15 metres shorter than Isabellas. To calculate how long Avery's jump rope is, we must subtract:
2.03 - 0.15 = 1.88
ANSWER:
The length of Avery's jump rope is 1.88 metres.
What is the constant of proportionality in the equation 2y=6x
Answers
The constant of proportionality in the equation 2y = 6x is [tex]\( \frac{1}{3} \)[/tex].
In the equation (2y = 6x), the constant of proportionality represents the relationship between x and y. To isolate it, divide both sides of the equation by (6x):
[tex]\[\frac{2y}{6x} = \frac{6x}{6x}\][/tex]
This simplifies to:
[tex]\[\frac{1}{3} = \frac{y}{x}\][/tex]
So, the constant of proportionality is [tex]\(1/3\)[/tex], indicating that for every unit increase in x, y increases by [tex]\(1/3\)[/tex].
What is the slope of a line that is perpendicular to the line shown?
(3, 3) (-3, -1)
Answers
Answer:
-3/2
Step-by-step explanation:
The line shown has slope ...
... (y2 -y1)/(x2 -x1) = (-1 -3)/(-3 -3) = -4/-6 = 2/3
The slope of the perpendicular line is the opposite of the reciprocal of this:
... -1/(2/3) = -3/2
Lionel is planning a one-day outing
Answers
Of the four choices given, which two, when written as a system, have a solution of (–4, 5)?
Answers
It's A i just took the test.
If John's car gets an average of 18 mpg and he fills his tank with 26.7 gallons of gas, how far can John travel on a tank of gas?
Answers
multiply 18 by 26.7 to get that answer
Answer:
480.6 miles.
Step-by-step explanation:
We have been given that John's car gets an average of 18 mpg. John fills his tank with 26.7 gallons of gas.
To find the distance traveled by John's car, we will multiply the distance covered per gallon by number of gallons in the tank.
[tex]\text{Total distance traveled by John's car}=\frac{18\text{ miles}}{\text{gallon}}\times \text{26.7 gallons}[/tex]
[tex]\text{Total distance traveled by John's car}=18\text{ miles}\times 26.7[/tex]
[tex]\text{Total distance traveled by John's car}=480.6\text{ miles}[/tex]
Therefore, John can travel 480.6 miles on a tank of gas.
Write the equation of the function graphed.
a)f(x) = |x + 4|
b)f(x) = |x – 4|
d)f(x) = |x| + 4
d)f(x) = |x| – 4
Answers
{that is, the last answer choice given.}.
_______________________________________________________
Note:
This graph contains the points: " (-4, 0) " ; and " (0, 4) " .
________________________________________________________
The last answer choice given: [D]: " f(x) = |x| – 4 " ; is the only answer choice containing BOTH points.
________________________________________________________
20-3(m-1)=-4 What is m??
Answers
20-3(m-1)=-4
20-3m+3=-4
23-3m=-4
-3m=-27
m=9
To check if this answer is correct, simply substitute m with 9 and you will get 9=9
If it's 10 A.M. Eastern Standard Time in New York City, what time is it in Los Angeles? A. 1:00 P.M. B. 9:00 A.M. C. 7:00 A.M D. 12:00 P.M.
Answers
The answer is C. 7:00 A.M.
~ TRUE BOSS
The table below shows the distance d(t) in feet that an object travels in t seconds:
t
(seconds)
d(t)
(feet)
3
126
4
224
5
350
6
504
What is the average rate of change of d(t) between 3 seconds and 5 seconds, and what does it represent?
Answers
Bonds from the city of Denver with a face value of $1,000 each are selling at 110.611. If Sean wishes to purchase three such bonds, how much can he expect to pay?
Answers
In this case, the bond quote is 110.611 which means that bond is traded at 110.611% of its face value.
If Sean wants to buy 3 bonds he needs to pay:
3*1000*110.611/100=$3318.33
Answer:
The answer is C-3,318.33
Step-by-step explanation:
just took the test homie
if the legs of a right triangle are 8 and 10, find the length of the hypotenuse
Answers
You need to use the Pythagorean theorem.
a² + b² = c²
8² + 10² = c²
64 + 100 = c²
164 = c²
c = √164
c = 2√41
Decimal is 12.8
Thus, the length of the hypotenuse is 12
Let me know if you have any misunderstanding about the answer!
As always, I am here to help!
We can use Pythagoras' Theorem to solve this problem. The Pythagorean Theorem:
a² + b² = c², where c is the hypotenuse.
Your two legs measure 8 and 10; substitute them into the theorem:
a² + b² = c²
8² + 10² = c²
64 + 100 = c²
164 = c²
12.81 ≈ c
Answer:
The length of the hypotenuse is approximately 12.81 units.
Samantha works at a nursery where she pots plants. It takes her 1 1/2 hours to pot 12 plants. How many plants can Samantha pot in 1 hour? (Please give me the equation you used.)
Please help fast!
Answers
You can use cross products to solve for x. Please see the attached picture for the steps. The answer is 8 plants.
Final answer:
Samantha can pot 8 plants in 1 hour.
Explanation:
To find out how many plants Samantha can pot in 1 hour, we need to determine how many plants she can pot in 1 1/2 hours and then convert that to 1 hour.
If Samantha can pot 12 plants in 1 1/2 hours, we can set up a proportion to find out how many plants she can pot in 1 hour:
12 plants / 1.5 hours = x plants / 1 hour
To solve for x, we can cross multiply:
12 * 1 = 1.5 * x
12 = 1.5x
To isolate x, divide both sides of the equation by 1.5:
x = 12 / 1.5
Simplifying the equation, we get:
x = 8
Therefore, Samantha can pot 8 plants in 1 hour.
4/15 divided by 8/25
Answers
- x - =. - simplify≈ -
15 8 120 6
multiply by the reciprocal to divide fractions
Scott viewed 72 classic cars at a car show.Of the cars he viewed,3/8 were sports cars. How many sports cars did Scott view
Answers
9x3=27
Therefore Scott viewed 27 sports cars.
Brandon sights a helicopter above a building that is 200 feet away at an angle of elevation of 30 degrees. To the nearest foot, How high above the ground the is the helicopter
Answers
We then have the following trigonometric relationship:
Tan (30) = y / 200
Clearing the value of and we have:
y = 200 * tan (30)
y = 115.4700538
Nearest foot:
y = 116 feet.
Answer:
the helicopter is 116 feet above the ground
A company pays its commercial agents 5% of any deal. What should the value of a deal be for the agent to earn $2000?
Answers
remember that 5% as a decimal is .05.Let x = value of deal
The equation would be .05x = $2000
Divide both sides by 0.05.
x=40,000
Answer:
The value of the deal should be $40,000.
Step-by-step explanation:
remember that 5% as a decimal is .05.Let x = value of deal
The equation would be .05x = $2000
Divide both sides by 0.05.
x=40,000
help help help !!!!!!!!
Answers
4 × 12/5 × 2
48/5 × 2
48/10
= 24/5
Keep, Change, Flip.
4/5 divided by 2/12.
Keep 4/5
Change division to multiplication.
Flip 2/12 to 12/2
4/5 x 12/2 =
24/5
the sum of the real numbers x and y is 15. their difference is 3. what is the value of xy?
Answers
x-y=3
the easiest way is the find out x and y first:
add up the two equations: 2x=18, x=9
plug x=9 in either equation, you get y=6
so xy=6*9=54
Another way to do it:
(x+y)²=15²=225 =>x²+2xy+y²=225
(x-y)²=3²=9 =>x²-2xy+y²=9
subtract one equation from the other: 4xy=225-9=216, xy=54
The value of xy is such that the sum of the real numbers x and y is 15 and their difference is 3 is 54.
What is a number system?The number system is a way to represent or express numbers.
A decimal number is a very common number that we use frequently.
Since the decimal number system employs ten digits from 0 to 9, it has a base of 10.
Any of the multiple sets of symbols and the guidelines for utilizing them to represent numbers are included in the Number System.
Given that sum of the x and y is 15
So,
x + y = 15
And difference
x - y = 3
By adding both equations,
2x = 18
x = 9 and y = 6
So their product 9 × 6 = 54.
Hence "The value of xy is such that the sum of the real numbers x and y is 15 and their difference is 3 is 54".
For more about the number system,
https://brainly.com/question/22046046
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This picture shows a feeding trough that is shaped like a right prism 5,4,6. list each face of the feeding trough . What is the surface area of the trough?
Please help
Answers
Two sides are 4 by 6.
Two sides are 5 by 6.
Since it's a feeding tough, I assume the top face is open and not there, so one of the six sides listed above is missing, but without a picture, I can't tell which one it is.
Answer:
Step-by-step explanation: