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122. 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 $sum straight n cubed space equals space left parenthesis sum straight n right parenthesis squared$

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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 $space space space 3 to the power of 4 minus 2 to the power of 4 space equals space 4.2 cubed space plus space 6.2 squared space plus space 4.2 space plus space 1 comma space space space space 4 to the power of 4 minus 3 to the power of 4 space equals space 4.3 cubed space plus space 6.3 squared space plus space 4.3 space plus 1$

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$left parenthesis straight n plus 1 right parenthesis to the power of 4 minus straight n to the power of 4 space equals space 4. straight n cubed plus 6. straight n squared plus 4. straight n plus 1$

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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Short Answer Type

123. Find the sum of n terms of the series whose nth term is

space space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed over denominator straight n end fraction.

128 Views

Long Answer Type

124.

Find the sum to n terms of the series:

1 . 2 . 4 + 2. 3 . 7 + 3. 4. 10 +..............

161 Views

Short Answer Type

125. Find the sum of the cubes of first n odd natural numbers. Or Sum the series to n terms l³ + 3³ + 5³ +..........

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126.

Find the sum to n terms the series $fraction numerator 1 over denominator 1.2 end fraction plus fraction numerator 1 over denominator 2.3 end fraction plus fraction numerator 1 over denominator 3.4 end fraction plus...........$ and deduce the sum to infinity.

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127.

Sum of the series n . 1 + (n - 1). 2 + (n - 2) . 3 + ........ + 1. n

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Long Answer Type

128. Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of these integers.

154 Views

Short Answer Type

129.

Show that $fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus...... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus...... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction$