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Giving this as to test the equation results in "False":

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But simply changing the upper level on the divider, it results in true

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What gives?

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Simplification is not done by default. Moreover, == (Equal) and === (SameQ) are not the same.

FullSimplify[(a == (a r - a)/(r - 1))]

True

You might not think you are expected to understand these subtleties of Wolfram Language, as Wolfram Alpha is meant as a human interpreter. However, by using === you are just misleading the interpreter. If you would use =, you also get the expected result.

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