Let's use algebra for this:
1st number: x
2nd number: (x+1)
3rd number: (x+2)
the 3 consecutive numbers add up to:
(x)+(x+1)+(x+2) = 3x+3
the 3 consecutive number multiply to:
(x) times (x+1) times (x+2) = (x^2+x)(x+2) = x^3 + 2x^2 + x^2 + 2x
= x^3 + 3x^2 + 2x
Now you tell me that the sum is 1/5th of their product which means that if we multiplied the sum by 5, we would get the product. Therefore, we can form the following equation:
sum * 5 = product
sum = 3x+3
product = x^3 + 3x^2 + 2x
5 * (3x+3) = x^3+ 3x^2 + 2x
15x + 15 = x^3 + 3x^2 + 2x
subtract fifteen from both sides
15x + 15 -15 = x^3 + 3x^2 + 2x -15
15x = x^3 + 3x^2 + 2x -15
subtract 15x from both sides
15x - 15x = x^3 + 3x^2 + 2x -15 - 15x
0 = x^3 + 3x^2 - 13x -15
now we have a cubic equation:
x^3 + 3x^2 - 13x -15 = 0
(x-3)(x+5)(x+1) = 0
x= 3, -5, -1
we now know our first consecutive number values: they can either be 3, -5 or -1.
I think you will only be looking for the positive values, so I'll only set out the solutions for 3.
first number = x = 3
second number = x+1 = 3 + 1 = 4
third number = x+2 = 3 + 2 = 5
therefore, our 3 consecutive numbers are 3,4,5.
I trust you can solve a cubic equation, so I didn't show how I solved it.. If you can't understand how I did it, email me back at atif.muhammad@gmail.com
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