SOLUTION: what three consecutive numbers have a sum which is one fifth of their product?

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Question 221062: what three consecutive numbers have a sum which is one fifth of their product?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
What three consecutive numbers have a sum which is one fifth of their product?

Step 1. Let n be one number

Step 2. Let n+1 and n+2 be the next two consecutive numbers.

Step 3. Then n+n+1+n+2=3n+3=3(n+1) be the sum of the three consecutive numbers.

Step 4. Then n*(n+1)*(n+2) be the product of the three consecutive numbers.

Step 5. Since the three consecutive numbers have a sum which is one fifth of their product, then

3%28n%2B1%29=n%28n%2B1%29%28n%2B2%29%2F5

Divide by n+1 to to sides of the equation

3%28n%2B1%29%2F%28n%2B1%29=n%28n%2B1%29%28n%2B2%29%2F%28n%2B1%29%2F5

Multiply by 5 to both sides of the equation to get rid of denominator

5%2A3=5%2A%28n%5E2%2B2n%29%2F5

15=n%5E2%2B2n

Subtract 15 from both sides to get a quadratic

n%5E2%2B2n-15=0

Step 6. Using the quadratic formula given as

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1, b=1, and c=-15

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B2x%2B-15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A1%2A-15=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+64+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+3
x%5B2%5D+=+%28-%282%29-sqrt%28+64+%29%29%2F2%5C1+=+-5

Quadratic expression 1x%5E2%2B2x%2B-15 can be factored:
1x%5E2%2B2x%2B-15+=+1%28x-3%29%2A%28x--5%29
Again, the answer is: 3, -5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-15+%29



n=+3, n%2B1=4, n%2B2=5

and

n=+-5, n%2B1=-4, n%2B2=-3

Step 7. ANSWER: The consecutive numbers are 3, 4, 5 AND -5, -4, and -3

I hope the above steps were helpful.

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Good luck in your studies!

Respectfully,
Dr J