Write the sum as
[tex]\displaystyle S = \sum_{k=1}^n \frac1{k(k+2)}[/tex]
Decompose the summand into partial fractions; that is, we look for constants a and b such that
[tex]\dfrac 1{k(k+2)} = \dfrac ak + \dfrac b{k+2}[/tex]
Condense the right side into a single fraction and set the numerators on both sides equal to one another:
[tex]\dfrac1{k(k+2)} = \dfrac{a(k+2)}{k(k+2)} + \dfrac{bk}{k(k+2)} = \dfrac{a(k+2)+bk}{k(k+2)} \\\\ \implies 1 = a(k+2) + bk = (a+b)k + 2a[/tex]
Then a + b = 0 and 2a = 1, so it follows that a = 1/2 and b = -1/2.
We then see that S is a telescoping sum (intermediate terms cancel and the overall sum collapses into a small number of terms):
[tex]\displaystyle S = \frac12 \sum_{k=1}^n \left(\frac1k - \frac1{k+2}\right) \\\\ S = \frac12 \left(1-\frac13 + \frac12 - \frac14 + \frac13 - \frac15 + \cdots + \frac1{n-2} - \frac1n + \frac1{n-1} - \frac1{n+1} + \frac1n - \frac1{n+2}\right) \\\\ S = \frac12 \left(1 + \frac12 - \frac1{n+1} - \frac1{n+2}\right) \\\\ \boxed{S = \frac{n(3n+5)}{4(n+1)(n+2)}}[/tex]
solve for x pls show your work
1/6x-1/8=11/12
Answer:
6 and 1/4
Step-by-step explanation:
1/6 x - 1/8 = 11/12
First, let's find the least common denominator (24) to make these fractions manageable to work with: 4/24 x - 3/24 = 22/24.
Next, add 3/24 to both sides: 4/24 x = 25/24.
We can cancel out the denominators at this point to get: 4x=25.
Divide both side by 4, and x = 25/4 = 6 and 1/4 or 6.25
Please help me fast this question
Answer:
64÷16+[12x{16÷(16-10)}]
64÷16+[12x{16÷6}]
64÷16+[12x2.66666666667]
64÷16+32
4+32
36
The measure of angle C __is
degrees, and the measure of angle D__ is
degrees.
Answer:
C=78
D=102
Step-by-step explanation:
co-interior angles are sepplementary
This clock shows the time that Abby starts school this morning.
What time does Abby start school?
Answer:
7:55am
Step-by-step explanation:
The hour hand is not quite at the eight so it would be 7. The minute hand is directly on the 11, from each number is 5 minutes multiplying 11×5 would be 55. therefore it being 7:55am which is morning.
Find the degree of 2b^9c^5
[tex] \large \mathfrak{Solution : }[/tex]
Degree of a polynomial is the highest exponential power in the expression.
i.e 9 here
so, degree of the given expression is 9
Help me with 19 and 20
19 find x
20 find y
Answer:
19) x = 48
20) y = 21
Step-by-step explanation:
19)
7x - 248 + x + 44 = 180
(because they're supplementary angles)
8x - 204 = 180
8x = 180 + 204
8x = 384
x = 48
20)
2y + 50 + 5y - 17 = 180
7y + 33 = 180
7y = 180 - 33
7y = 147
y = 21
D is the midpoint of RJ⎯⎯⎯⎯. R has coordinates (−4,7), and D has coordinates (2,−1). Identify the coordinates of J.
Answer:
[tex](8,-9)[/tex]
Step-by-step explanation:
If the x-coordinate of R is 6 units away from D's x-coordinate, then so will J's x-coordinate. If the y-coordinate of R is 8 units away from D's y-coordinate, then so will J's y-coordinate.
Side note/Extra information:
The line would be [tex]y=\frac{-4}{3} x+\frac{5}{3}[/tex]
please help, this is geometry
Answer:
Step-by-step explanation:
r = 3
angle = 50
Formula: Length = (angle/360) * 2 pi * r
Length = (50/360) * 2 * 3.14 * 3
Length = 0.13889 * 2 * 3.14 * 3
Length = 0.13899 * 6 * 3.14
Length = 2.6167
y= -23 +9
y = 42 - 3
PLEASE HELP
Answer:
y = -14
y = 39
Step-by-step explanation:
Step-by-step explanation:
are u sure this is possible
are these simultaneous equations
The table represents the equation y = 2 - 4x
х
y
Use the drop-down menus to complete the statements
The x-values are the
The y-values are the
The missing value in the table for x = -1 is y = 1
V
-2
10
-1
0
2
1
-2
-6
N
It
Answer:
the x-values are the inputs
the y-values are the outputs
the missing value in the table for x = -1 is y = 6
Step-by-step explanation:
Which is greater 55.5% or 3/5
Answer:
3/5
Step-by-step explanation:
3 divided by 5 is 0.6, which translates to 60%.
55.5% < 60%
Answer:
3/5 is the greater .
I hope it will help you. Have a great day ahead.
What integer is equivalent to 25 3/2
Add using the number line.
−0.5+(−1.7)
Select the location on the number line to plot the sum.
Answer:
-2.2
Step-by-step explanation:
-5 -4.5 -4 -3.5 -3 -2.5 -2.2 -2 -1.5 -1 -0.5 0
Does the quotient of 4554 divided by 9 have a remainder? How can you tell without dividing?
Answer:
No; Because 9 is a factor of 4554. By divisibility rules sum of digits of 4554 is 18, which is a number divisible by 9. Hence 4554 is divisible by 9.
Step-by-step explanation:
:)
-17 + 20 + -3 = ?
also please try to explain it if u can :3
Answer:
000000000000000
Step-by-step explanation:
no explanation
A scientist uses 3 gallon of liquid for every 6 hours he uses (blank) of a gallon each hour he works
Answer:
2 gallons per hour
Step-by-step explanation:
6 divided by 3
NEED HELP ASAP!
Carolina and Patricia received their Christmas bonus and decided to use this weekend to buy their Christmas gifts .
Carolina decided to take advantage of the Great Winter Sale in Liverpool that offers discounts of 50% + 20%.
Patricia went to the mall and took advantage of the boutique discount that offers discounts of 70%.
They both bought 3 items with a total listing price of $120. At the end of their purchase rounds, they will have coffee with their friend Cristina and tell her about the offers they found. Based on their experiences, Cristina must decide where she will do her shopping.
Analyze this situation and think,
Where would you recommend Cristina to do her shopping? What is the final price paid on each establishment?
Applying what you have seen in your math classes, what where your calculations and how you can relate it to composition of functions.
Answer:
I would recommend Christina to do her shopping at any of those 2 places because they both have a discount of 70%. The final price paid on each establishment was $36.
120*0.7 = 84
120 - 84 = 36
Step-by-step explanation:
Carolina has a discount of 50%+20% which is the same as a discount of 70%, and Patricia had a discount of also 70%, so they both had the same discounts.
What number can replace c
Answer:
3
Step-by-step explanation:
If a block has a volume of 216 cm³ in length and width of sides respectively 12 cm and 6 cm. What is the height of the cuboid?
Step-by-step explanation:
I hope it helped U
Javonie started a dog walking business over the summer. He walked 8 dogs and earned $160 in July.
He walked 9 dogs and earned $180 in August.
a. How much money did he earn over the summer?
b. How much did he charge per dog in July?
*Determine what you need to find.
*Set up the problem.
*Solve.
*Check.
Answer:
a) $340
b) $20
Step-by-step explanation:
a) in the summer (July and August) he earned 160 + 180 = $340
b) in July he earned 160 ÷ 8 = $ 20
Javonie earned $340 over the summer and he charged $20 per dog in July.
The amount of money he earned over the summer is the sum of the money he earned in July and the sum of the money he earned in August
Total amount earned in the summer = amount earned in July + amount earned in August
$160 + $180 = $340
The amount he charged per dog in July can be determined by dividing the total amount earned in July by the total number of dogs he walked.
Amount charged per dog = amount earned in July / total dogs worked
$160 / 8 = $20
A similar question was solved here : https://brainly.com/question/15219860?referrer=searchResults
Fractions, decimals, and percents are all used to describe a
part of something. Which of the 3 types do you think is most
commonly used? Why do you suppose this is true?
Answer:
Step-by-step explanation:
I think that Percents are used very commonly all around us every day. It gives us a better vision in our mind of the quantity than decimals or fractions. We see Percents everywhere. In the shopping mall for discounts, math homework and work. So overall, percents are most commonly used.
Hope that helps!
How much simple interest is earned on $6,000 at an interest rate of 4.52% in 4 1/2 years?
Answer:
1220.4
Step-by-step explanation:
S.I.=P×R×T÷100
=6000×4.52×4 1/2÷100
=6000×4.52×9÷2×100
=60×4.52×9÷2
=30×4.52×9
=1220.4
Find the sum of the given arithmetic series. 14 + 28 + 42 + 56 + ... + 280
14 + 28 + 42 + 56 + … 280
= 14 (1 + 2 + 3 + 4 + … + 20)
= 14 × 20 × 21 / 2
= 2940
where we use the well-known identity,
[tex]\displaystyle \sum_{i=1}^ni = 1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}2[/tex]
Write 7.12 as a mixed number in simplest form.
Work Shown:
7.12 = 7 + 0.12
7.12 = 7 + (12/100)
7.12 = 7 + (3/25)
7.12 = 7 & 3/25
The notation 7 & 3/25 means we have the whole part 7 and fractional part 3/25.
Using the Triangle Sum Theorem. Determine the angle measure
of <1.
38
31
(78 and 31 degrees)
Answer:
71 degrees
Step-by-step explanation:
Hi there!
The sum of the interior angles of a triangle is always 180 degrees.
Knowing this, to solve for the measure of angle 1, subtract 78 and 31 degrees from 180:
180-78-31
= 71
Therefore, angle 1 measure 71 degrees.
I hope this helps!
Help if you don’t mind I would appreciate it
Answer:
1/5
Step-by-step explanation:
the answer would be 1/5 because on the chart it showed:
5-2
10-3
15-4
20-5
which changed by 1/5
Collect 5 large numbers and 5 small numbers related in life and express them as exponents. Sample- Distance of the Earth from the moon
384400 km = 3.844 × 10⁵ km
Answer:
Exponential notation is a powerful way to express repeated
multiplication of the same number. Specifically, powers of 10
express very large and very small numbers in a manner which is
convenient to read, write and compare.
• For any non-zero integer a, a–m 1
m a
=
• Laws of exponents are
(a) am × an = am+n
(b) am ÷ an = am–n
(c) (am)
n = amn
(d) am × bm = (ab)
m
(e) a° = 1, where a ≠ 0
(f)
m m
m
a a
b b
=
• Numbers can be expressed in expanded form by using exponents.
• Very large and very small numbers can be expressed in standard
form.
• Standard form is also called scientific notation form.
Step-by-step explanation:
Example 1 : Multiplicative inverse of 27 is
(a) 2–7 (b) 72 (c) – 27 (d) – 27
Solution : The Correct answer is (a).
Example 2 : The human body has about 100 billion cells. This number
can be written in exponential form as
(a) 10–11 (b) 1011 (c) 109 (d) 10–9
Solution : The correct answer is (b).
In examples 3 to 5, fill in the blanks to make the statements true.
Example 3 :
4
4 5 (– 4) ______ 4
× =
Solution : 54
Example 4 : ( ) ( ) 2 3 –3 –2 2 3 _____ × =
Solution : 6–6
Example 5 : The distance between earth and sun is 150 million
kilometres which can be written in exponential form as
_______.
Solution : 1.5 × 108 km
In examples 6 and 7, state whether the statements are true (T) or false (F):
Example 6 : Very small numbers can be expressed in standard form
using positive exponents.
Solution : False.
Example 7 : (–10) × (–10) × (–10) × (–10) = 10–4
Solution : False.
Step-by-step explanation:
Simplify the expression
Answer:
[tex] {( - 4)}^{5} \times {( - 4)}^{7} \\ = {( - 4)}^{5 + 7} \\ = {( - 4)}^{12} \\ = 16777216 \\ thank \: you[/tex]
-5v+3(-8v-2)=3v-38
Plz show the steps I don’t understand
PQRS is a rectangle.
The length of PM and QM are shown in the diagram
what is the value of X?
Answer:
diagram pls
then I can answer this question
