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I have the question:

$$1 + 8 + 27 + \dots + n^3 = \frac {n^2(n+1)^2}n$$

But doesn't this fail the base case? If we plug in a $1$ on both sides of the problem we get $1 = 4$. Which is not true. Does that mean we can't prove this further?

Any help would be extremely appreciated!

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Damian M
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2 Answers2

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The denominator should be $4$. Obviously, the right-hand side needs to be a quartic.

J.G.
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  • That's what I thought too. But this was the problem that was given. Is there anyway to prove this as it is? – Damian M Apr 26 '18 at 20:14
  • you already disproved it...@DamianM it's false for $n=1$ – user577215664 Apr 26 '18 at 20:15
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    @DamianM You can't prove something that's wrong, but you can find all $n$ for which the equation holds. Whether it was intended as such an exercise or is an identity with a misprint I don't know. – J.G. Apr 26 '18 at 20:16
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A mathematical proof can be:

I - The problem has a solution, and is determined (or the statement is true);

II - The problem has solutions, and are general and undetermined (or the statement is true for $n$);

III - The problem has no solution (or the statement is false)

Any of the above would be accepted.

You just proved that the statement above is false (III) by assigning $n = 1$.

You could ask your teacher if the intended question is actually wrong (and therefore the denominator on the right side should be 4) or if that was exactly what the question was about, and that is, mathematical proofs of any kind.

Lucas Hipólito
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