Permutations and Combinations

Question

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

Solution

L.H.S = $fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus......... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus..... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator sum straight n left parenthesis straight n plus 1 right parenthesis squared over denominator sum straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator sum straight n left parenthesis straight n squared plus 2 straight n plus 1 right parenthesis over denominator sum straight n squared left parenthesis straight n plus 1 right parenthesis end fraction$
= $fraction numerator sum left parenthesis straight n cubed plus 2 straight n squared plus straight n right parenthesis over denominator sum left parenthesis straight n cubed plus straight n squared right parenthesis end fraction space equals space fraction numerator sum straight n cubed plus 2 sum straight n squared plus sum straight n over denominator sum straight n cubed plus sum straight n squared end fraction space equals space fraction numerator begin display style fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction end style plus begin display style fraction numerator 2 straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction end style plus begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style over denominator begin display style fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction end style plus begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction end style end fraction$
= $fraction numerator begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style plus begin display style fraction numerator 2 left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 end fraction end style plus begin display style 1 over 1 end style close square brackets over denominator begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style plus begin display style fraction numerator 2 straight n plus 1 over denominator 3 end fraction end style close square brackets end fraction space equals space fraction numerator 3 straight n squared plus 3 straight n plus 8 straight n plus 4 plus 6 over denominator 3 straight n squared plus 3 straight n plus 4 straight n plus 2 end fraction$
= $space space space space space fraction numerator 3 straight n squared plus 11 straight n plus 10 over denominator 3 straight n squared plus 7 straight n plus 2 end fraction space equals space fraction numerator 3 straight n squared plus 6 straight n plus 5 straight n plus 10 over denominator 3 straight n squared plus 6 straight n plus straight n plus 2 end fraction equals space fraction numerator 3 straight n left parenthesis straight n plus 2 right parenthesis space plus space 5 left parenthesis straight n plus 2 right parenthesis over denominator 3 straight n left parenthesis straight n plus 2 right parenthesis plus 1 left parenthesis straight n plus 2 right parenthesis end fraction$
= $fraction numerator left parenthesis straight n plus 2 right parenthesis space left parenthesis 3 straight n plus 5 right parenthesis over denominator left parenthesis straight n plus 2 right parenthesis space left parenthesis 3 straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction$

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Permutations And Combinations

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Determine K, so that K + 2, 4K – 6 and 3K – 2 are three consecutive terms of an A.P.

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