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
Solve for n:
n
---- = 1
20
A. n = 23
B. n = 15
C. n = 20
D. n = 32
Answers
Answer:
The answer is C
Step-by-step explanation:
[tex]\frac{n}{20}*20 = 1 * 20[/tex]
1 x 20 = 20
Answer:
C) n = 20
Step-by-step explanation:
hope it helps!
When using estimates to make decisions,
A. It is important to always use a 95% confidence level
B. It is important to remember that you will never achieve certainty
C. It is important to calculate those estimates to at least six decimal places
D. It is important to use a 100% confidence level so as to achieve certainty
E. All of the above
F. None of the above
Answers
Answer:
C.
Explanation: I hope it's help you
The use of estimates to make the decisions It is important to calculate those estimates to at least six decimal places.
We have given the options that are
A. It is important to always use a 95% confidence level
B. It is important to remember that you will never achieve certainty
C. It is important to calculate those estimates to at least six decimal places.
D. It is important to use a 100% confidence level so as to achieve certainty.
E. All of the above
F. None of the above
We have to determine the correct decision.
What are the estimating values?
Any of numerous procedures used to calculate the value of some property of a population from observations of a sample drawn.
That is we are estimating the value up to 6 decimal places
Therefore option C is correct.
It is important to calculate those estimates to at least six decimal places.
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Write the value of (2 +√ 2) (2 -√3)
Answers
[tex]\left(2 + \sqrt 2 \right)\left(2-\sqrt 3 \right)\\\\=4- 2\sqrt 3 + 2 \sqrt 2 - \sqrt 6\\\\=0.91484.....[/tex]
Which is the graph of 4x – 3y = 12?
Answers
Answer:
it is the last graph
Step-by-step explanation:
A point located at (5, 3) is translated down 4 units and right 2 units. What are the coordinates of the image?
(9, 5)
(1, 5)
(7, -1)
(3, 7)
Answers
Answer:
(7, -1)
Step-by-step explanation:
Let's remember our key words:
Down/up means to translate the y axis.
Right/left means to translate the x axis.
Now that we know what to do, let's look at our equation.
----------------------------------------------------------------------------------------------------------
A point located at (5, 3) is translated down 4 units and right 2 units.
Let's work on the x axis first.
It says that you are translating to the right. That means your value is going higher, so you go to the right 2. 5+2=7
So now that we have our x, (7), let's rewrite our equation. (7, 3)
----------------------------------------------------------------------------------------------------------
Next, let's do y.
The points are translating down. That means our value will lessen.
Since we are going down on our y axis by 4, we will subtract our current y coordinate with 4. 3-4=-1.
Now we know our y, so let's rewrite our equation with our new y. (7, -1).
----------------------------------------------------------------------------------------------------------
Our final coordinates are (7, -1).
----------------------------------------------------------------------------------------------------------
I hope this helps!
-No one
consider the equation 2 x +9 y = -18 A line parallel to the above line would have a slope of_________
A line perpendicular to the above line would have a slope of ________________
Answers
A line parallel to the above line would have a slope of -2/9
A line perpendicular to the above line would have a slope of 9/2
Step-by-step explanation:
2x+9y=-18
=>y=-2/9 x-2
An amount of R351 is made up with 114 coins (R2 coins and R5 coins only). How many R2 coins are there? How many R5 coin are there?
Answers
Answer:
41
Step-by-step explanation:
For this, you have to make two equations;
One that calculates the amount of coins
and one that calculates the amount of R
I'm going to set
R2 = x
R5 = y
To make this a bit easier to look at
So to do the number of coins, you want to do this:
x + y = 114
This is saying that the total amount of R2 coins plus the total amount of R5 coins will add up to 114
Then, you want to calculate the worth of them.
2(x) + 5(y) = 351
This is saying that for ever R2 coin there is, it will increase the total by 2, and every R5, it will increase the total by 5. When you add these together, it will come out to 351.
Then you have to solve the system of equations:
x + y = 114
2x + 5y = 351
-2 * (x + y = 114)
2x + 5y = 351
-2x + -2y = -228
2x + 5y = 351
3y = 123
y = 41
Since y is the amount of R5 coins, we know we have 41 R5 coins.
So the answer is
41
The product of an irrational number and an irrational number is irrational?
Always true
Sometimes true
Never true
Answers
[tex]========================================[/tex]
is the product of an irrational number and an irrational number always irrational?
We can easily check that by just multiplying some irrational numbers on our calculator.
Let's take
[tex]\pmb{\sqrt{2} }[/tex] times [tex]\pmb{\sqrt{5} }[/tex]
Multiply:-
The result is an irrational number.
Let's try to multiply π times itself.
Once again, we have an [tex]\pmb{irrational}[/tex] number.
Hence, the answer is:-
[tex]\pmb{Always~True}[/tex]
[tex]=========================================[/tex]
note:-Hope everything is clear; if you need any more explanation, kindly let me know.One tube of paint covers 1,400 square inches of ramp. Based on this, how many tubes of paint will be needed to paint the ramp?
Use your answer from the question above to help you answer this question.
my answer above was
2,905 square inches
Answers
The tubes of paint that would be needed to paint the ramp with an area of 2905 inches is 3 tubes of paint
What is division?Division is a mathematical operation that is used to determine the quotient of a number. The sign used to represent division is ÷.
What is the number of tubes of paints needed?2905 / 1400 = 2.075 tubes of paint
To learn more about division, please check: https://brainly.com/question/194007
Answer:
you really go to the cca
Step-by-step explanation:
you really go to the cca
Tim had 85$ to spend on 6 books. After buying them he had 13$. How much did each book cost. Write using two step equations
Answers
Answer:
he spent 12 dollars on each book
Step-by-step explanation:
minus 13 from 85 you get 72 and then divide 72 by 6
Answer:
12$ each
Step-by-step explanation:
Tim had 85$ to spend on 6 books.
After buying them he had 13$.
85$ - 13$ = 72$How much did each book cost.
72$ / 6books = 12$ each bookFind the mean of the data.
55,64,58,43,49,67 Please!!
Answers
Help help help math math math
Answers
Answer:
adjacent
Step-by-step explanation:
share the same side
Answer:
adjacent
Step-by-step explanation:
Label each point on the number line with the correct value.
Click each dot on the image to select an answer. Choices:
Answers
Answer:
dot A is 0.62 dot B is 7/9 and dot C is 10/9
Step-by-step explanation:
What is the first term of the arithmetic series?
a1 =
143
⇒ 8
What is the 20th term of the arithmetic series?
a20 =
⇒ 46
How many stitches are in the first 20 rows of the shawl?
⇒ 540 stitches
Answers
I don't think this is wrong
Which of the following is described below?
This object is infinitely small. It is
comprised of a single location.
B. Line
A. Point
D. Plane
C. Ray
Answers
Answer:
A. Point
Step-by-step explanation:
A point is infinitely small and comprised of a single location.
hope this helps!! p.s. i really need brainliest :)
The object that is infinitely small among the option is a point. Option A is correct.
What is standard measurement?All over the world, most things are in use as of common quantity, so people all over the world agree that they have to follow a common system of measurement and that the common system of measurement is called standard measurement.
Here,
The object that is infinitely small among the option is a point. Because there is no instrument that can measure the point. Since a point is nothing but a demonized dot. While line, ray, or plane are infinitely long and their measure can be possible.
Thus, the object that is infinitely small among the option is a point. Option A is correct.
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Find the length of each arc shown in red. Leave your answer in terms of pi. 110 4in
Answers
The length of the arc shown in red is
Step-by-step explanation:
step 1
Find the circumference of the circle The circumference is equal to we have substitute
step 2
Find the length of the arc in red Remember that the length of the circumference subtends a central angle of 360 degrees so by proportion find the length of the arc by a central angle of 110 degrees Simplify
if julie make 18 baskets how many baskets does kari make how many baskets do they make in all
Answers
Answer:
18x
Step-by-step explanation:
j=18
k=x
18xor18
I do not understand this question
show how you know, thank you :)
~giving brainliest!~
Answers
Answer:
V≈180.96
Step-by-step explanation:
16. What is the area of the shaded region in terms of Tr? [20 points]
Answers
= πr2
= π(5)2
=25 π
Area of the bigger circle inside
= π x(3)2
= 9 π
Area of the smaller circle inside
= π x (2)2
= 4 π
Area of shaded region
= 25π - 9π - 4π
= 12π square unit
the original price: $75 and percent of discount :30% what’s the
Answers
Answer:
The discount is 22.50 and the final price $52.50
Step-by-step explanation:
Discount = Original Price x Discount %/100
Discount = 75 × 30/100
Discount = 75 x 0.3
You save = $22.50
Final Price = Original Price - Discount
Final Price = 75 - 22.5
Final Price = $52.50
Simplify Sinx + cosx/ cos²x - sin²x
Answers
[tex]\dfrac{\sin x + \cos x}{\cos^2 x - \sin^2 x}\\ \\\\=\dfrac{\sin x + \cos x}{(\cos x + \sin x )\\\\(\cos x - \sin x)}\\\\\\=\dfrac{1}{\cos x - \sin x}[/tex]
Is quadrilateral ABCD angle is equals to 60 degrees Angle B is equals to 80 degrees and angle C is equals to 120° find angle d
Answers
Angle D=360-(60+80+120)
=100 degrees
please solve for x please use statement reason
Answers
AE || BC
===》 DCB angle = DAE angle
AC /
__________________________________
Thus :
DCB angle = x
And also we know that the submission of interior angles of any triangle must equals to 180 degrees so :
In BCD triangle :
B + C + D = 180
x + x + D = 180
D = 180 - 2x
__________________________________
In the other hand we know that :
BDC angle + BDA angle = 180
because they make a straight line together so ;
180 - 2x + 50 = 180
- 2x = 180 - 230
- 2x = - 50
x = 25
And we're done...
Have a great time ♡
Help help math math help
Answers
Answer:
The answer to this equation is 2
Step-by-step explanation:
x-2 = 3(2-x) \4
x-2= 6 -3x/4
cross multiply
4(x-2)= 6-3x
4x-8 = 6-3
collect like terms
4x+3x = 6+8
7x = 14
divide both sides by 7
7x/7= 14/7
x = 2
The problem is write you answer using ONLY positive exponents
Answers
Answer:
z^10/z^5 = z^5
Step-by-step explanation:
if it is divide, you minus, so z^10/z^5 is z^5.
The nisure below is made of 2 rectangular prisms. What is the volume of this figure? 5 m 4 m 2 m 9 m 5 m
Answers
Answer: Find the volume of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively. Therefore, the volume of a rectangular prism is 192 cm 3.
Find the average of this set of numbers.16, 20, 30, 30
A) 14
B) 15
C) 25
D) none of the above
Answers
Answer:
D) none of the above
Step-by-step explanation:
To find an average, you add up all of the numbers and divide by how many numbers there were. for example, in this equation, the sum of all of the numbers was 96. then I divided 96 by 4(the amount of numbers shown)
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
PLSS HELP ME
12/12 + 12/12 + 7/12=_____
Answers
2 7/12
This is because 12/12 is equal to 1/1 or 1. So 12/12 + 12/12 = 2.
2 + 7/12= 2 7/12
Answer:
31
-----
12
Step-by-step explanation:
12 12 7 31
---- + ------- + ----- = -------
12 12 12 12
Add the numerators together (top number) and keep the denominator the same (bottom number)
I hope this helps!
What is the domain for the following function?
y=-x+1
x²+x-6
X-
O A. {X-1
O B. (x*-3
O c. {# -3;x # 2)
OD. x+0)