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]
10.
a. Explain how to add two integers with the same sign.
b. Explain how to add two integers with different signs.
Answer:
a) To add integers having the same sign, keep the same sign and add the absolute value of each number. To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest. Subtract an integer by adding its opposite.
b) To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest. Subtract an integer by adding its opposite.
Step-by-step explanation:
Mark me brainliest if it helps you .
Find the value of x
Answer:
Step-by-step explanation:
The exterior angle (shown) is equal to the two remote interior angles ( the ones marked with values in x).
5x + 2 + 5x = 11x - 11 Combine the terms on the left.
10x + 2 = 11x - 11 Add 11 to both sides
10x + 2 + 11 = 11x
10x + 13 = 11x Subtract 10x from both sides.
13 = x
The actual distance between A and B is 960 metres.
a) Work out the scale on the map. Give your answer in the form 1:n
b) Work out the actual distance between A and C in kilometres. Your final line must say, AC = ... km
Answer:
Step-by-step explanation:
a. 960 / 4 = 240 for 1 grid
1/2 cm = 24000
1 : 48000
b. A to C
AC² = 3² + 6²
AC² = 9 + 36
AC²= 432 + 1728
AC² = 2160
AC = √2160 km
Which expression is equivalent to (5³) ‐²? (4 points)
1) 1 over 5 to the 3rd
2) negative 1 over 5 times 5 times 5 times 5 times 5 times 5 times
3) 5³
4) 1 over 5 times 5 times 5 times 5 times 5 times 5 times
Answer:
1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸
Step-by-step explanation:
exponential law
Answer:
1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2 = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2 = 3⁶
3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]
= (-3)3+4
= (-3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)
= m5+3
= m⁸
Step-by-step explanation:
Line segment XY has endpoints X(5, 7) and Y(– 3, 3). Find the equation for the perpendicular bisector of line segment XY.
Answer: The equation for the perpendicular bisector of line segment xy is .
Explanation:
It is given that the line segment xy has endpoints x(5,7) and y(-3,3).
The bisector divides the line segment xy in two equal parts, so the bisector must be passing through the midpoint of xy.
Midpoint of two points (x_1,y_1) and (x_2,y_2) is calculated as,
Midpoint of xy is,
So, the perpendicular bisector must be passing through the point (1,5).
The slope of line passing through the point (x_1,y_1) and (x_2,y_2) is calculated as,
Slope of line segment is . Therefore the slope of perpendicular bisector is -2 because the the product of slopes of two perpendicular lines is always -1.
The line passing through the point (x_1,y_1) with slope m is defined as,
Bisector passing through the point (1,5) with slope 2.
Therefore, the equation for the perpendicular bisector of line segment xy is .
I got confused I add or multiply 30 3 times? Or am i wrong
Answer:
30=3x
so divide 30 by 3 and you'll get x
Answer:
10
Step-by-step explanation:
You equate. The two angles are equal. The figure is called vertically opposite.
Vertically opposite angles are equal.
3x = 30 Now you divide by 3
3x/3 = 30/3
x = 10
And 10 is your answer.
Julio is paid 1.4 times his normal hourly rate for each hour he works over 30 hours in a
week. Last week he worked 35 hours and earned $436.60. Write and solve an equation to
find Julio's normal hourly rate, r. Explain how you know that your answer is reasonable.
Answer:
His normal hourly rate is $11.80 / hr
Step-by-step explanation:
Is it important to study geography? Why or why not?
Step-by-step explanation:
to some, its important because without it, they might not understand the surroundings e.g pilot,e.t.c while others, it is not important because they don't seriously need to understand their surroundings e.g doctor
Please help me and tysm i’m advance!!
Answer:
[tex]r = \sqrt{\frac{3V}{'pi'h}}[/tex]
replace the 'pi' in that equation to the symbol pi, π
Step-by-step explanation:
70x2 ? A 10 B 24 C 100 D 140
Answer:
140
Step-by-step explanation:
[tex] \mathrm{Answer} : [/tex]
70 × 2 = 140therefore, correct answer is D. 140
a^15/a^5=a^10 true or false
Answer:
true
Step-by-step explanation:
a^15 ÷ a^5 = a^10
a÷a = a
15- 5 =10
give brainliest please I need it to level up
Answer:
True
Step-by-step explanation:
Using the rule of exponents
[tex]\frac{a^{m} }{a^{n} }[/tex] = [tex]a^{(m-n)}[/tex] , then
[tex]\frac{a^{15} }{a^{5} }[/tex]
= [tex]a^{(15-5)}[/tex]
= [tex]a^{10}[/tex]
i^32=......................
Find the smallest value of k, such that 16k is a perfect cube.
Answer:
k = 4
Step-by-step explanation:
16k = 16(4) = 64 and
64 = 4 × 4 × 4
[tex]\sqrt[3]{64}[/tex]
= [tex]\sqrt[3]{4^{3} }[/tex]
= 4
How would I plot and label the numbers 6.5 & 8.5 on a number line to write a inequality using the symbol “>” to compare the two numbers
Answer:
To label 6.5 on a number line, put the plot between the 6 and 7.
Same thing goes for 8.5, except the plot would go between the 8 and 9.
8.5 > 6.5
Step-by-step explanation:
1.) Since 8.5 and 6.5 are decimals, they are not a whole number which is why they go right in the middle of the two numbers they are behind and in front of.
2.) > means greater than, and 6.5 is smaller than 8.5 therefore we use the greater than sign, pointed towards the 8.5.
I hope this helps! :)
what do you mean by a reference point?
Step-by-step explanation:
A reference point is a place or object used for comparison to determine if something is in motion.
solve the inequality.
|7x-5| < -3
Answer:
no solution
Step-by-step explanation:
3<7x−5<−3
Answer:
No solution
Step-by-step explanation:
The absolute value function is always positive or 0
What's the circumference of a
circle with a radius of 6 inches?
Use 3.14
C = [?] inches
Answer:
37.7
Step-by-step explanation:
Answer:
C=2πr=2·π·6≈37.69911in
The math club made pies to sell for a Pi Day fundraiser. The cafeteria donated 12 pies. All of the pies were cut into 8 slices, and the math club sold all 208 slices. How many pies did the math club make?
Answer:
They made 26 pies.
Step-by-step explanation: brainliest please
3/2[(58-10^2)÷(√16 +3)]
Answer:
- 1/4 or -0.25 is the answer
If i round 706 to the nearest 10 and to the nearest 100, i get the same answer
The slope of a graphed line is 2/3 and the y-intercept is (0, 4). What is the slope-intercept equation of the line? A.y=-4x+2/3 B.y=2/3x+4 C.y= -2/3x+4 D.y=4x-2/3
Answer:
B.y=2x C.y=1/2x D.y=-x 2.
Step-by-step explanation:
HELP PLEASE I NEED FASS ANSWERS TO ALL IF CAN
Answer:
8: 12 ,5
9 : 18, 7
10 : 17,5
But Not Sure
What does the word Tangent mean ?
Answer:
ur mom XD :/
Step-by-step explanation:
Answer:
A straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point.
a square and arectangle have the same perimeter calculate the area of the rectangle if the side of the 60 cm and the lenght of the rectangle is 80
the food service industry is projected to grow from 13.1 million to nearly 14.4 million employees by 2023.calculate the percent increase
Answer: 9.92% increase
Step-by-step explanation:
(14.4MM-13.1MM)/(13.1MM) = 0.099237 0.0992 to 3 sig figs
9.92% increase
Please answer all four geometry questions and show work... I will mark you brainliest and give you five stars
Answer: for the first one it is the ll because on the first one it measures 80 degrees for both angles and so for the the second just that it's 90 degrees
For the second 2 questions the are parallel because they don't join together but go straight until infinite.
Step-by-step explanation:
10.
Which is the graph of the exponential function y = 9(3)x?
Answer:
The answer is B
Step-by-step explanation:
Answer:
B is right
Step-by-step explanation:
I got 100 on my test, give the other person brainliest.
What is the bionominal expansion of (x+2y)^7
Answer:
Choose the 8th row of Pascal's triangle to get the sequence:
1
,
7
,
21
,
35
,
35
,
21
,
7
,
1
Write the powers of
2
in ascending order up to
2
7
as a sequence:
1
,
2
,
4
,
8
,
16
,
32
,
64
,
128
Multiply the two sequences together to get:
1
,
14
,
84
,
280
,
560
,
672
,
448
,
128
Then:
(
x
+
2
y
)
7
=
x
7
+
14
x
6
y
+
84
x
5
y
2
+
280
x
4
y
3
+
560
x
3
y
4
+
672
x
2
y
5
+
448
x
y
6
+
128
y
7
Given the expression below, which statement is true about the
parentheses?
10 + (26 - 4)*2
1.The parentheses indicate that 10 * 26 should be simplified last
2.The parentheses indicate that 10 * 26 should be simplified first.
3.The parentheses indicate that 26 - 4 should be simplified first.
4.The parentheses indicate that 10 + 26 should be simplified first.
please help i have ZERO knowledge in math
Answer:
your correct answer is 3 no first ,
how do i divide easy
Answer:
Depending on what difficulty of division you are doing, if it's simple division such as 120 ÷ 12, you can solve that by remembering your multiplication table. If you want to know how to solve long division, tell me and I'll extend my answer.
