Permutations and Combinations

Permutations and Combinations

Question

Find the sum of n terms of the series 3 + 7 + 13 + 21 + 31 +.......

Answer

Let  straight S subscript straight n space equals space 3 plus 7 plus 13 plus 21 plus 31 plus......... space plus space straight T subscript straight n minus 1 end subscript space plus space straight T subscript straight n
Re-writing it as straight S subscript straight n space equals space 3 plus 7 plus 13 plus 21 plus..... plus straight T subscript straight n minus 2 end subscript space plus space straight T subscript straight n minus 1 end subscript plus straight T subscript straight n
Subtracting,     0 space equals space 3 plus left square bracket 4 plus 6 plus 8 plus 10 plus 1..... to space left parenthesis straight n space minus space 1 right parenthesis space terms right square bracket space minus space straight T subscript straight n.
∴                   straight T subscript straight n space equals space 3 plus fraction numerator straight n minus 1 over denominator 2 end fraction. space left parenthesis 2 straight n plus 4 right parenthesis space equals space 3 space plus space left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n plus 2 right parenthesis
or                space space straight T subscript straight n space equals space straight n squared plus straight n plus 1
∴                  straight S subscript straight n space equals space sum from straight k equals 1 to straight n of straight T subscript straight k space equals space sum from straight k equals 1 to straight n of left parenthesis straight k squared plus straight k plus 1 right parenthesis space equals space sum from straight k equals 1 to straight n of straight k squared space plus space sum from straight k equals 1 to straight n of space straight k space plus space sum from straight k equals 1 to straight n of 1
or                  straight S subscript straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction plus straight n space equals space straight n open square brackets fraction numerator left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus fraction numerator left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction plus 1 close square brackets
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                          equals space space straight n over 6 left square bracket 2 straight n squared plus 6 straight n plus 10 right square bracket space equals space straight n over 3 left parenthesis straight n squared plus 3 straight n plus 5 right parenthesis

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