How can i prove this property to be true for all values of n, using mathematical induction.
ps: spam/wrong answers will be reported and blocked.
Answers
Proof -
So, in the first part we'll verify by taking n = 1.
[tex] \implies \: 1 = {1}^{2} = \frac{1(1 + 1)(2 + 1)}{6} [/tex]
[tex] \implies{ \frac{1(2)(3)}{6} }[/tex]
[tex]\implies{ 1}[/tex]
Therefore, it is true for the first part.
In the second part we will assume that,
[tex] \: { {1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} = \frac{k(k + 1)(2k + 1)}{6} }[/tex]
and we will prove that,
[tex]\sf{ \: { {1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k + 1)(k + 1 + 1) \{2(k + 1) + 1\}}{6}}}[/tex]
[tex] \: {{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k + 1)(k + 2) (2k + 3)}{6}}[/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{k (k + 1) (2k + 1) }{6} + \frac{(k + 1) ^{2} }{6} [/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{k(k+1)(2k+1)+6(k+1)^ 2 }{6} [/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k+1)\{k(2k+1)+6(k+1)\} }{6}[/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k+1)(2k^2 +k+6k+6) }{6} [/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k+1)(2k^2+7k+6) }{6} [/tex]
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k+1)(k+2)(2k+3) }{6} [/tex]
Henceforth, by using the principle of mathematical induction 1²+2² +3²+....+n² = n(n+1)(2n+1)/ 6 for all positive integers n.
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Related Questions
What is 12.985 rounded to 2 decimal place?
Answers
12.985 rounded to 2 decimal places is 12.99
Answer:
Step-by-step explanation:
Step 1:
Identify the digit in the hundredths place, as it is the 2nd decimal place. The 8 is the digit in the hundredths place.
Step 2:
Look at the digit in the thousandths place and its value is 5 thousandths, so increase the digit in the hundredth's place to 9.
Step 3:
The new number is 12.99 rounded to the 2nd decimal place.
What is the dividend yield on Stock A that sells at
$15/share, when Company A pays a quarterly
dividend of $0.20 per share?
dividend yield = [?] %
Give your answer as a percent rounded to the
nearest tenth
Hint: Dividend Yield
annual dividend
share price
Answers
Answer:
5.3%
Step-by-step explanation:
The annual dividend is 4 times the quarterly dividend, so is $0.80 per share. The yield is the ratio of that to the share price:
yield = $0.80/$15.00 ≈ 0.05333... ≈ 5.3%
The dividend yield is 5.3%.
Sara and her friends are enjoying a bag of candy-coated chocolates. Just for fun, they remove 80 pieces of candy from the bag to check their color. The outcomes are shown in the table.
Candy Color Number of Candies
brown or orange 25
green or yellow 28
red or blue 27
Which statement is true based on the results in the table?
A.
All of the outcomes are equally likely because their relative frequencies are similar.
B.
All of the outcomes are equally likely because the relative frequency of picking brown or orange candies is greater than that of picking green or yellow.
C.
All of the outcomes are not equally likely because there is a lot of variation in their relative frequencies.
D.
All of the outcomes are not equally likely because the relative frequency of picking green or yellow candies is the highest.
E.
All of the outcomes are not equally likely because the relative frequency of picking red or blue candies is less than that of picking green or yellow.
Answers
Answer:
A
Step-by-step explanation:
There doesn't appear to be any particular color that has several more than the other choices.
Answer:
Option E
Step-by-step explanation:
If she picked them randomly than the 80 she picked are a pretty good representation of the candies in the bag, we know that there are different amounts of each candy based on Sara's numbers. According to her results, there are more green or yellow candies than brown or orange so the odds are there are more green or yellow candies in the bag. Since there are different numbers of each candy, the odds of her picking a certain color is different than the others, so the outcomes are not equally likely. The answeres can now be C,D, or E. I think it's option E because to me it seems like the most accurate of the three.
length of a rectangle is three times its width. if the length of the rectangle is 64 in, find the width and length
Answers
Answer:
Length: 64
Width = 21 1/3
Step-by-step explanation:
Width = Length 64 * 1/3 Width
Width = 21.3333333 or 21 1/3
Answer:
[tex]l=64 in[/tex]
[tex]w=21.3333in[/tex]
Step-by-step explanation:
[tex]l=3w[/tex]
[tex]\frac{64}{3} =21.3333[/tex]
The average age for the 62 observed milk cows as they first produced
Answers
Answer:
32
Step-by-step explanation:
as i thing that brother thank you
Complete or justify each step. 2(2h + 2)2(3h + 5) - 12 Distributive Property 4h + 4 < 6h-2 Simplify (combine like terms). 4h+ 4- 6h 6h-2-6h - 2h + 4<-2 simplify-2h<-6 simplify h>3
Answers
Answer:
7[tex]\geq[/tex][tex]\frac{h}{14}[/tex]
Step-by-step explanation:
Mark brainliest
Triangle ABC is isosceles.
Triangle A B C is shown. The lengths of sides A C and A B are congruent. Angle C A B is (x + 5) degrees. Angle A B C is (3 x) degrees.
What is the measure of angle C?
25°
30°
60°
75°
Answers
Answer: 75⁰°
Answer: 75⁰°Step-by-step explanation:
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25plug 25 in for x to find the angle measure
Answer: 75⁰°Step-by-step explanation:Using the Base Angles Theorem, we can conclude that <ABC and <ACB are congruent. The angles of a triangle add up to 180 degrees so we can write this equation to solve for x:x + 5+ 3x + 3x = 180simplify7x + 5 = 180subtract 5 from both sides7x = 175divide each side by 7x = 25plug 25 in for x to find the angle measure3(25) = 75
F(4g2-9)=n i need to solve for f
Answers
Answer:
Step-by-step explanation:
What is the next time: 8:30 A.M., 10:00 A.M., 11:30 A.M., _______? A) 1:00 P.M. B) 1:00 A.M. C) 1:30 P.M. D) 1:30 A.M.
Answers
Answer:
A 1 PM
Step-by-step explanation:
Becasue it grows by an hour and a half each time
Answer: Easy. A. 1 pm.
.
PLEASE HELP!!! I"LL MARK AS BRAINLIEST!!!A local water park has two types of season passes. Plan A costs a one-time fee of $139 for admission plus $8 for parking every trip. Plan B costs a one-time fee of $43 for parking plus $24 for admission every trip. How many visits must a person make for plan A and plan B to be equal in value?
Answers
Answer:
6 visits
Step-by-step explanation:
Plan A: 139+8x
Plan B: 43+24x
Set the equations equal to each other.
[tex]139 + 8x = 43 + 24x[/tex]
Subtract 43 from both sides.
[tex]96 + 8x = 24x[/tex]
Subtract 8x from both sides.
[tex]96 = 16x[/tex]
Divide both sides by 16.
[tex]x = 6[/tex]
A rectangular room is 21 feet wide. It is
three times as long as it is wide. What is
the perimeter of the room?
Answers
Answer:
I think the answer is 7
Step-by-step explanation:
7 multipled my 3 equals 21.
What is the volume of a sphere with a diameter of 50 cm
Answers
since we know the diameter is 50, then its radius is half that or 25.
[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=25 \end{cases}\implies \begin{array}{llll} V=\cfrac{4\pi (25)^3}{3}\implies V=\cfrac{62500\pi }{3} \\\\\\ V\approx 65449.85~cm^3 \end{array}[/tex]
5
5
What is the equation of the line that is parallel to the
given line and has an x-intercept of -3?
3
2
+
(3, 1)
-5 -4 -3
2
=
3
4
5
х
-11 2
| (0, 1)
-2.
O y=x+3
O y=x+2
O y=-*x+3
O y=-x+2
-3
-5
Answers
Answer:
b
Step-by-step explanation:
The solution is, the slope-intercept form equation is, y = -4x -6
What is a straight line?A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.
here, we have,
The given segment has a rise if 1 for a run of 4, so a slope of ...
m = rise/run = 1/4
The desired perpendicular has a slope that is the negative reciprocal of this:
m = -1/(1/4) = -4
A point that has a rise of -4 for a run of 1 from the given midpoint is ...
(-1, -2) +(1, -4) = (0, -6) . . . . . . . the y-intercept of the bisector
So, our perpendicular bisector has a slope of m=-4 and a y-intercept of b=-6. Putting these in the slope-intercept form equation, we find the line to be ...
y = mx +b
y = -4x -6
To learn more on Equation 0f Straight-Line click:
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#SPJ7
Find the Sum.
29, 8 67
+ 14, 938
Answers
Answer:
44805
Step-by-step explanation:
i just added them together lol
Answer:
the answer is 44805
Step-by-step explanation:
just add them together
help pls i really need help
Answers
Answer:
See belowStep-by-step explanation:
We see the segment of length of 3 is divided by 3 and each of units divided by 4.
The number of purple rectangles is 12.
This can be expressed as:
Number of fourths in 3, this is same as 123 ÷ 1/4 = 12, same as aboveThe other options are not matching this model. They all end up with part of the line segment.
Three of them result in 3/4 and one results in 4.
Which of the following choices represents the average of other four?
A)4. B)-3. C). -4. D). -17. E). 5
Answers
Answer:
B
Step-by-step explanation:
Let's add some numbers together and divide by the numbers of numbers. The answer should be among these.
4 + 5 + -4 + -3 = 2
4/2 = 2.
-17 + -4 + 4 + 5 = -12
-12/4 = -3.
The answer is B.
Help help help help
Answers
Answer:
h=3/2. V=3(root)3
Step-by-step explanation:
I will mark the first person to answer my question brainliest
Answers
Answer:
1. The length is longer.
2. The length is longer.
Step-by-step explanation:
1. 7/10 = 0.70 > 0.67. Here, the length is greater than the width.
2. 9/10 of a yard is .9 * 36, or 32.4 inches. Here, the length is greater than the width.
What is the explict rule for the geometric sequence? 9,6,4, 8/3 Enter your answer in the box. an=
Answers
Answer:
an = 9(⅔)^(n-1)
Step-by-step explanation:
The formula for explicit is: An = a1(r)^(n-1)
For a period of time, an island’s population grows at a rate proportional to its population. If the growth rate is 4.5% per year and the current population is 1500, what will the population be 6 years from now
Answers
The population of the island with growth rate of 4.5% per year and the current population is 1500 is 1953 after 6 years.
How to solve an exponential function
Let y represent the population after x years.
Given that the growth rate is 4.5% per year and the current population is 1500.
Hence a = 1500, b = 100% + 4.5% = 104.5% = 1.045
y = 1500(1.045)ˣ
After 6 years:
y = 1500(1.045)⁶ = 1953
The population of the island with growth rate of 4.5% per year and the current population is 1500 is 1953 after 6 years.
Find out more on exponential function at: https://brainly.com/question/12940982
hey guy can you please explain this to me
Answers
Answer:
D) 2
Step-by-step explanation:
it think it's D because the size of the triangle was doubled
2 was matched to 4 and
1 was matched to 2
Can someone please help me with this
Answers
Answer:
Vertex: (-1 , 50)
Axis of Symmetry: x = -1
y - intercepts: (0 , 48)
Find f(x+2) if f(x)=(5x^2)-2x+3
Answers
Answer:
Use the given functions to set up and simplify f(x+2).
5x2+18x+19 This is the Answer if using The Function Operation.
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10x−2 This is the Answer if you want to find the Derivative
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I don't know if my answer is right if it's wrong I am sorry.
Anyways have an amazing day!
may anyone please help me with this too , thank you so much
Answers
7) 7bc cubed
8) 6x to the power 8
which one is greater 7 pounds or 70 ounces
Answers
Answer:
7 pounds is greater
Step-by-step explanation:
1 lb = 16 oz
7 lb = 112 oz
70 oz is 70 oz
Answer:
7 pounds is greater
Step-by-step explanation:
1 lb = 16 oz
7 lb = 112 oz
70 oz is 70 oz
You should be able to figure out the answer
Classement pays médaille d'or de cyclisme moyennes 70 %pays...
Answers
Answer:
Classement pays médaille d'or de cyclisme moyennes 70 %paga...
Step-by-step explanation:
Write these numbers in order of size. Start with the smallest number. 3007 4435 399 4011 3333
Answers
Hey there!
Let's write all these numbers in order.
First, all numbers here are 4-digit numbers, except 1 (it's a 3-digit number)
So the smallest number is the 3-digit number.
399
Now, what's next?
let's check the thousands place (the first digit in the 4-digit numbers)
We see 3, 3, 4 and 4.
3 is smaller than 4.
Now we have 2 numbers: 3007 and 3333.
Well, which one's smaller? that's right, 3007.
So we have
399, 3007.
The third number is 3333 (the thousands place is 3)
so we have
399, 3007, 3333
Now, we have 2 numbers left, and they have a 4 in their thousands place.
The numbers are: 4011 and 4435.
Well, which one's smaller? that's right, 4011, and the greatest number is 4435.
Hence, we have
[tex]\boxed{\boxed{\bold{399,~3007,~3333,~4011,~4435}}}[/tex]
Hope everything is clear.
Let me know if you have any questions!
#LearningDoesn'tEnd
:-)
50 Points! A section of wall for a mural is shaped like a parallelogram with a base length of 7 feet and a height of 11 feet. A can of paint will cover 20 square feet.
nvm it was 4
Answers
Area = length x height:
Area = 7 x 11 = 77 square feet.
divide area of wall by area per can:
77/20 = 3.85 cans
they will need to buy 4 cans
A doctor conducts a survey with a random sample of his patients, measuring their cholesterol levels in millimolar per liter.
The data measurements for 14 patients are given below. Assuming the population standard deviation is o = 1.3 and that
the population is normally distributed, compute a 90% confidence interval for the mean cholesterol levels of his patients.
Round your answers to two decimal places and use ascending order.
Answers
Using the z-distribution, as we have the standard deviation for the population, it is found that the 90% confidence interval for the mean cholesterol levels of his patients is (5.59, 6.73).
What is a t-distribution confidence interval?The confidence interval is:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.z is the critical value.n is the sample size.[tex]\sigma[/tex] is the standard deviation for the sample.In this problem, we have a 90% confidence level, hence[tex]\alpha = 0.90[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.9}{2} = 0.95[/tex], so the critical value is z = 1.645.
Researching on the internet, the other parameters are given by:
[tex]\overline{x} = 6.16, \sigma = 1.3, n = 14[/tex]
Hence:
[tex]\overline{x} - z\frac{\sigma}{\sqrt{n}} = 6.16 - 1.645\frac{1.3}{\sqrt{14}} = 5.59[/tex]
[tex]\overline{x} + z\frac{\sigma}{\sqrt{n}} = 6.16 + 1.645\frac{1.3}{\sqrt{14}} = 6.73[/tex]
The 90% confidence interval for the mean cholesterol levels of his patients is (5.59, 6.73).
More can be learned about the z-distribution at https://brainly.com/question/25890103
If the difference of 3 / 4 and 1 / 6 of a number is 7, find the number and also verify your answer.