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Prove That (1^(3))/( 1) + (1^(3) + 2^(3)...

Prove That `(1^(3))/( 1) + (1^(3) + 2^(3))/(1+3) + (1^(3) + 2^(3) + 3^(3))/( 1 + 3+ 5)` + . . . upto n terms `= (n)/(24) [2n^(2) + 9n + 13]`

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(1^(2) )/( 1) + (1^(2) + 2^(2) )/(1+2) + (1^(2) + 2^(2) + 3^(2) )/( 1+ 2+ 3)+ …. + n terms =

1^(2) + 3^(2) + 5^(2) + …. upto n terms =

S_(n) = 1^(3) + 2^(3) + 3^(3) + …... + n^(3) and T_(n) = 1+ 2 + 3+ 4…...n

1^(3) + 1^(2) + 1+2^(3) + 2^(2) + 2+3^(2) + 3^(2) + 3+3… 3n terms =

1+ ((1)/(3) + (1)/(3^2) ) + (( 1)/( 3^3) + (1)/( 3^4) + (1)/( 3^5) ) + .... sum of the terms in the n^( th) bracket=

(1)/(1.3) + (1)/(3.5) + (1)/(5.7) + …. (n-3) terms

Prove that (1)/(1!(n-1)!) + (1)/(3!(n-3)!)+ (1)/(5!(n-5)!) + …….= (2^(n-1))/(n!)