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Fed [463]
2 years ago
8

Prove that 2n^3 + 3n^2 + n is divisible by 6 for every integer n > 1 by mathematical induction. (7 marks)

Mathematics
1 answer:
notka56 [123]2 years ago
5 0

Answer:

The answer is True

Step-by-step explanation:

A <em>mathematical induction</em> consists in only 2 steps:  

<u>First step</u>: Show the proposition is true for the first one valid integer number.

<u>Second step</u>: Show that if any one is true then the next one is true  

Finally, if first step and second step are true, then the complete proposition is true.

So, given 2*n^3+3*n^2+n

First step: using and replacing n=2 (the first valid integer number >1)

2*(2)^3 +3*(2)^2+2=30

\frac{30}{6} =5

As the result is an integer number, so the first step is true.

Second step: using any next number, n+1, let it replace

2*(n+1)^3+3*(n+1)^2+(n+1)\\2*(n^3+3*n^2+3*n+1)+3*(n^2+2*n+1)^2+(n+1)\\2*n^3+6*n^2+6*n+2+3*n^2+6*n+3+n+1\\(2*n^3+3*n^2+n)+(6*n^2+12*n+6)

As the First step is true, we know that

2*n^3+3*n^2+n=6*k,

So let it replace in the previous expression

6*k+6*(n^2+2*n+1)\\6*[k+(n^2+2*n+1)]

Finally

\frac{6*[k+(n^2+2*n+1)]}{6} =k+(n^2+2*n+1)

where the last expression is an integer number

So the second step is true, and the complete proposition is True

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Step-by-step explanation:

Squared with 8 and 6 imagine 6 parcel and 8 sides.

It minus's to equal the equation 2.

3 0
3 years ago
Item 9 It costs $68 for 16 square feet of flooring. How much does it cost for 12 square feet of flooring?
Travka [436]
To solve you divide 68 by 16 and that gives you the price per square foot so that is $4.25, Now times 12 by that and the answer will be $51 dollars.
3 0
3 years ago
-6(-4x + 3) = 2(10x + 3)
Naya [18.7K]

Answer:

x = -6/11

Step-by-step explanation:

-6(-4x + 3) = 2(10x + 3)

-6 * -4x = -24x

-6 * 3 = -18

2 * 10x = 20x

2 * 3 = 6

-24x - 18 = 20x + 6

-20x           -20x

-44x - 18 = 6

+18           +18

-44x = 24

-------   ------

 -44      -44

24/-44 = -24/44

-24/44 = -6/22

x = -6/11

8 0
3 years ago
What is the volume of the triangular prism? Round to the nearest tenth.
kramer

Answer:

the volume of the triangular prism is 1,185.6  in³

Step-by-step explanation:

To find the volume of these triangular prism, we will, first, find the area of the triangle then multiply by the length

Area of triangle = \frac{1}{2} × b×h

From the diagram ,the height of the triangle  is given to be 10.4 in and the base is given to be 12 in.

We will now proceed to insert the values into the formula;

                 Area of triangle  = \frac{1}{2} × b×h

                                              = \frac{1}{2} × 12×10.4

                                             

                                                = \frac{1}{2} × 124.8

                                                 =62.4

Area of triangle is 62.4 in²

From the diagram is given to be 19 in

Volume of triangular prism = area of triangular prism × l

                                              =62.4 in²   ×    19 in

                                              =1,185.6  in³

Therefore the volume of the triangular prism is 1,185.6  in³

                                           

                           

5 0
2 years ago
Find the factors of x^3 + z^3
bixtya [17]
So to factor
factor out a x+z
(x+z)(x^2-xz+z^2)
the factors are
x+z and x^2-xz+z^2
5 0
3 years ago
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