Permutations and Combinations

Permutations and Combinations

Question

Find the sum of the cubes of first n natural numbers.  Or

Evaluate  space space sum straight n cubed space space or space space space sum from straight k space equals 1 to straight n of space straight k cubed  Also, show that sum straight n cubed space equals space left parenthesis sum straight n right parenthesis squared

Answer

Consider an identity left parenthesis straight x plus 1 right parenthesis to the power of 4 space minus space straight x to the power of 4 space equals space 4 straight x cubed plus 6 straight x squared plus 4 straight x plus 1
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                          space space space 3 to the power of 4 minus 2 to the power of 4 space equals space 4.2 cubed space plus space 6.2 squared space plus space 4.2 space plus space 1 comma space space space space 4 to the power of 4 minus 3 to the power of 4 space equals space 4.3 cubed space plus space 6.3 squared space plus space 4.3 space plus 1
                             .............................................................
                             .............................................................
                   left parenthesis straight n plus 1 right parenthesis to the power of 4 minus straight n to the power of 4 space equals space 4. straight n cubed plus 6. straight n squared plus 4. straight n plus 1
Adding vertically, we get
       space space space left parenthesis straight n plus 1 right parenthesis to the power of 4 minus 1 to the power of 4 space equals space 4 left parenthesis 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed right parenthesis space plus space 6 left parenthesis 1 squared plus 2 squared plus 3 squared plus....... plus straight n squared right parenthesis
                                                                                  plus space space 4 left parenthesis 1 plus 2 plus 3 plus...... plus straight n right parenthesis space plus space straight n
straight n to the power of 4 plus 4 straight n cubed plus 6 straight n squared plus 4 straight n plus 1 minus 1 space equals space 4 sum straight n cubed plus space 6 sum straight n squared plus 4 sum straight n plus straight n

rightwards double arrow     4 sum straight n cubed space equals space straight n to the power of 4 plus 4 straight n cubed plus 6 straight n squared plus 3 straight n minus 6 sum straight n squared minus 4 sum straight n
                    equals straight n to the power of 4 plus 4 straight n cubed plus 6 straight n squared plus 3 straight n space minus space fraction numerator 6 straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction space minus space fraction numerator 4 straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
                    equals space straight n to the power of 4 plus 4 straight n cubed plus 6 straight n squared plus 3 straight n minus straight n left parenthesis straight n plus 1 right parenthesis space left parenthesis 2 straight n plus 1 right parenthesis minus 2 straight n left parenthesis straight n plus 1 right parenthesis
                     equals straight n left square bracket straight n cubed plus 4 straight n squared plus 6 straight n plus 3 minus left parenthesis 2 straight n squared plus straight n plus 2 straight n plus 1 right parenthesis minus 2 straight n minus 2]
rightwards double arrow         sum straight n cubed space equals space straight n over 4 left square bracket straight n cubed plus 4 straight n squared plus 6 straight n plus 3 minus 2 straight n squared minus 3 straight n minus 1 minus 2 straight n minus 2 right square bracket
                       equals straight n over 4 left square bracket straight n cubed plus 2 straight n squared plus straight n right square bracket space equals space straight n squared over 4 left parenthesis straight n squared plus 2 straight n plus 1 right parenthesis space equals space fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction space equals space open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets squared space equals space left parenthesis sum straight n right parenthesis squared
∴   sum from straight k equals 1 to straight n of straight k cubed or  sum straight n cubed space space space equals space 1 cubed plus 2 cubed plus 3 cubed plus........ plus straight n cubed space equals space fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator straight n end fraction space equals space left parenthesis sum straight n right parenthesis squared
                      

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