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USPshnik [31]
3 years ago
14

Fred and Frank play a game. Fred picks an even number X and asks Frank to add all even numbers between 0 and X (inclusive) as fa

st as possible. After some rounds Frank found a quick way to do it. He simply multiplies X/2 by (X/2 + 1). Explain why it always works.
Mathematics
1 answer:
TiliK225 [7]3 years ago
8 0

Answer:

The even numbers between 0 and X represents an arithmetic sequence with a common difference of 2

The rule of arithmetic sequence = a + d(n - 1)

Where a is the first term and n is the number of terms

So, for the even numbers between 0 and X

The first term = a = 0

d = 2

So, we need to find n at the last term which is X

∴ X = 0 + 2 ( n -1 )

∴ n - 1 = X/2

∴ n = X/2 + 1

The sum of the arithmetic sequence = (n/2) × (2a + (n−1)d)

Substitute with a and d and X

So, the sum = (n/2) * (2*0 + (n−1)*2)

                   = (n/2) * ((n−1)*2)

                   = n(n-1)

                   = (X/2 + 1) * (X/2)

                   = X/2 by (X/2 + 1)

So, The quick way to add all even numbers between 0 and X always works.

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PLEASE HELP!
Kitty [74]
You can reword the two equations as:

-5x-y=15 (Divide original value by 3)
-2x+6y=6

Then use elimination to find x:

-30x-6y=90 (Multiply by 6 to get y values to be same to cancel out)
-2x+6y=6

You're left with:

-32x=96. Which can then be solved to find x which is -3.

Then plug back in 

-2x+6y=6

Now to: -2(-3)+6y=6.

Which reduces to 6+6y=6. So y=0.



To graph them, just reword the equations (yes once again) so that y is in front.

y=-5x-15 and y=(1/3)x+1






8 0
3 years ago
M=-3/7 ; (-5,6) what is the equation of the line
exis [7]
Y - 6 = -3/7(x + 5)

y - 42/7 = -3/7x - 15/7

y = -3/7x + 27/7
6 0
3 years ago
Find the length of the hypotenuse of a right triangle with legs of lengths 12 ft and 16 ft answer
Finger [1]
Use Pythagorus theorem: 

a²+b²=c²
(12)²+(16)²=c²
400=c²
√400=c 
20=c 

Therefore, the hypotenuse would be 20 ft. 

Hope I helped :) 
8 0
2 years ago
I need help on this quick
Pachacha [2.7K]

Answer:

The hypotenuse to the nearest tenth is 8.1

Step-by-step explanation:

We can use the Pythagorean theorem to solve

a^2 + b^2 = c^2 where a and b are the legs and c is the hypotenuse

Let a be the x leg and b be the y leg

a = 7 units

b= 4 units

7^2 + 4^2 = c^2

49+ 16 = c^2

65 = c^2

Take the square root of each side

sqrt(65) = sqrt(c^2)

8.062257748 =c

To the nearest tenth

8.1 =c

7 0
3 years ago
Answer this one please
Fittoniya [83]

Answer:

105 cm²

Step-by-step explanation:

We can see that the area at the square at the bottom, since one side is 5, is 5²=25. Then, we know that the height of the triangles is 8, the base is 5, and there are 4 of them, so ((5*8)/2)*4=80. Add these up to get 105 as our total

4 0
3 years ago
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