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Y_Kistochka [10]
3 years ago
12

Show that n(n+1)(n+2) is divisible by 6Solution(​

Mathematics
1 answer:
vladimir2022 [97]3 years ago
6 0

Proof by induction

Base case:

n=1: 1*2*3=6 is obviously divisible by six.

Assumption: For every n>1 n(n+1)(n+2) is divisible by 6.

For n+1:

(n+1)(n+2)(n+3)=

(n(n+1)(n+2)+3(n+1)(n+2))

We have assumed that n(n+1)(n+2) is divisble by 6.

We now only need to prove that 3(n+1)(n+2) is divisible by 6.

If 3(n+1)(n+2) is divisible by 6, then (n+1)(n+2) must be divisible by 2.

The "cool" part about this proof.

Since n is a natural number greater than 1 we can say the following:

If n is an odd number, then n+1 is even, then n+1 is divisible by 2 thus (n+1)(n+2) is divisible by 2,so we have proved what we wanted.

If n is an even number" then n+2 is even, then n+1 is divisible by 2 thus (n+1)(n+2) is divisible by 2,so we have proved what we wanted.

Therefore by using the method of mathematical induction we proved that for every natural number n, n(n+1)(n+2) is divisible by 6. QED.

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Answer:

(-3, -1)

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6 0
3 years ago
What is the surface area of a cube if it has one side that is 40 cm
belka [17]
The answer to that problem is 9600cm. each side of surface area will be the same multiplied together
5 0
3 years ago
The diameter of a wheel of a lorry is 150cm. If the lorry travels 376.8m, how many times does the wheel rotate?
tresset_1 [31]
<h3>Answer:  79 full rotations</h3>

====================================================

Explanation:

150 cm = 150/100 = 1.5 m

The wheel has a diameter of 1.5 meters

The circumference of the wheel is

C = pi*d

C = pi*1.5

C = 1.5pi

C = 4.71238898038469

I'm using my calculator's stored version of pi to get the most accuracy.

Then we divide the 376.8 over the circumference found

(376.8)/(4.71238898038469) = 79.9594434093682

Despite being very close to 80, we must round down to 79 because we don't have enough to get that full 80th rotation. In other words, we have 79 full rotations and then some change leftover.

Though for the sake of simplicity, I can see how it's useful to say "about 80 rotations" if 79 seems a bit clunky. I'll stick with 79 however. Let me know if your teacher instructs otherwise.

4 0
2 years ago
there ar 2.54 centimeters in 1 inch. how many cantimeter are in 1 foot? in 1 yard explain you reasoning
stellarik [79]
2.54 * 12=30.48, so 30.48 in one foot, then multiply that by three for a yard, so for a yard it is 91.44 .
4 0
3 years ago
1. What is a perfect square? 2. What is the square root of 4x²? 3. What is the square root of 25? 4. Factor 4x² + 20x + 25.
Ulleksa [173]

Answer:

Step-by-step explanation:

1) A perfect square is a whole number which is a product of a smaller whole number and itself. Examples of perfect squares are

4(2 × 2)

9(3 × 3)

16(4 × 4)

25(5 × 5)

36(6 × 6)

2) Square root of 4x² is 2x(product of square root of 4 and square root of x²)

3) square of 25 is 5

4) 4x² + 20x + 25

The general formula for solving quadratic equations is expressed as

x = [- b ± √(b² - 4ac)]/2a

From the equation given,

a = 4

b = 20

c = 25

Therefore,

x = [- 20 ± √(20² - 4 × 4 × 25)]/2 × 4

x = [- 20 ± √(400 - 400)]/8

x = [- 20 ± 0]/8

x = - 20/8

x = - 2.5

3 0
3 years ago
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