Find Formulas for n Dimensions 

Derive a formula for the region measure of an -dimensional unit ball.

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X Table[RegionMeasure[Ball[n]], {n, 10}]

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X Table[RegionMeasure[Ball[n]], {n, 10}]; FindSequenceFunction[%, n]

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Derive a formula for the region measure of an -dimensional unit simplex.

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X Table[RegionMeasure[Simplex[n]], {n, 10}]

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X Table[RegionMeasure[Simplex[n]], {n, 10}]; FindSequenceFunction[%, n] // FullSimplify

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Derive a formula for the region measure of an -dimensional unit sphere.

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X Table[RegionMeasure[Sphere[n]], {n, 10}]

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X Table[RegionMeasure[Sphere[n]], {n, 10}]; FindSequenceFunction[%, n] // FullSimplify

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Derive a formula for the integral of over an -dimensional unit ball.

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X Table[Integrate[Sum[Indexed[x, i]^2, {i, n}], x \[Element] Ball[n]], {n, 10}]

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X Table[Integrate[Sum[Indexed[x, i]^2, {i, n}], x \[Element] Ball[n]], {n, 10}]; FindSequenceFunction[%, n]

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