# Find Formulas for n Dimensions

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

 In[1]:= XTable[RegionMeasure[Ball[n]], {n, 10}]
 Out[1]=
 In[2]:= XTable[RegionMeasure[Ball[n]], {n, 10}]; FindSequenceFunction[%, n]
 Out[2]=

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

 In[3]:= XTable[RegionMeasure[Simplex[n]], {n, 10}]
 Out[3]=
 In[4]:= XTable[RegionMeasure[Simplex[n]], {n, 10}]; FindSequenceFunction[%, n] // FullSimplify
 Out[4]=

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

 In[5]:= XTable[RegionMeasure[Sphere[n]], {n, 10}]
 Out[5]=
 In[6]:= XTable[RegionMeasure[Sphere[n]], {n, 10}]; FindSequenceFunction[%, n] // FullSimplify
 Out[6]=

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

 In[7]:= XTable[Integrate[Sum[Indexed[x, i]^2, {i, n}], x \[Element] Ball[n]], {n, 10}]
 Out[7]=
 In[8]:= XTable[Integrate[Sum[Indexed[x, i]^2, {i, n}], x \[Element] Ball[n]], {n, 10}]; FindSequenceFunction[%, n]
 Out[8]=

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