Permutations and Combinations

Permutations and Combinations

Question

Find the sum to n terms of the series:

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

Answer

Let Tn be the nth terms of the series
       Tn = [nth terms of 1, 2, 3, .......] cross times [ nth terms of 2, 3, 4, ......] cross times [nth term of 4, 7, 10, ........]
 
          = [ 1 + (n -1) 1] [2 + (n - 1)1] [4 + (n - 1)3]
          = (1 + n - 1) (2 + n - 1) (4 + 3n - 3) = n(n + 1) (3n + 1)
rightwards double arrow     Tn  = 3n3 + 4n2 + n
Let   Sn denote the sum of n terms of the series:
 ∴   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 3 straight k cubed plus 4 straight k squared plus straight k right parenthesis space equals space 3 sum from straight k equals 1 to straight n of straight k cubed plus 4 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 straight k

                       equals fraction numerator 3 straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction plus fraction numerator 4 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 space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator 3 straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction plus fraction numerator 4 left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 end fraction plus 1 close square brackets
 = space space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction open square brackets fraction numerator 9 straight n squared plus 9 straight n plus 16 straight n plus 8 plus 6 over denominator 6 end fraction close square brackets space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 12 end fraction left square bracket 9 straight n squared plus 25 straight n plus 14 right square bracket
fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 12 end fraction left square bracket 9 straight n squared plus 18 straight n plus 7 straight n plus 14 right square bracket space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 12 end fraction space left square bracket 9 straight n left parenthesis straight n plus 2 right parenthesis space plus space 7 left parenthesis straight n plus 2 right parenthesis right square bracket
space space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis left parenthesis 9 straight n plus 7 right parenthesis over denominator 12 end fraction

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