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Math Help - Logic Proof

  1. #1
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    Logic Proof

    hey does anyone know how to solve this?

    Prove 1x2+2x3+3x4+...+n(n+1)=1/3n(n+1)(n+2) for each n of the element of natural numbers. x = multiplication

    thanks...
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  2. #2
    Moo
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    Hello,

    You're asked the sum :

    \sum_{k=1}^n k(k+1) \ = \ \sum_{k=1}^n k^2 + \sum_{k=1}^n k

    Do you know the general formulae for these sums ? You should have learnt it...i think ^^
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  3. #3
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    Hello, TheMikego!

    Prove: . 1\cdot2+2\cdot3+3\cdot4+ \hdots + n(n+1)\:=\:\frac{n(n+1)(n+2)}{3}\;\;\text{ for all }n \in N
    How about an inductive proof?

    We have: . S(n) \:=\:1\cdot2+2\cdot3+ 3\cdot4 + 4\cdot5 + \hdots + n(n+1)\;=\;\frac{n(n+1)(n+2)}{3}


    Verify S(1)\!:\;\;1\cdot 2 \:=\:\frac{1\cdot2\cdot3}{3} \:=\:2\quad\hdots\quad\text{True!}


    Assume S(k)\!:\;\;1\cdot2+2\cdot3+3\cdot4 + \hdots + k(k+1) \;=\;\frac{k(k+1)(k+2)}{3}



    Add (k+1)(k+2) to both sides:

    . . \underbrace{1\cdot2+2\cdot3+3\cdot4+\hdots+(k+1)(k  +2)}_{\text{This is the left side of }S(k+1)} \;=\;\frac{k(k+1)(k+2)}{3} + (k+1)(k+2)


    The right side is: . \frac{k(k+1)(k+2)}{3} +\frac{{\color{blue}3}(k+1)(k+2)}{{\color{blue}3}}

    . . \text{Factor: }\;\frac{(k+1)(k+2)}{3}\,[k+3] \;=\;\underbrace{\frac{(k+1)(k+2)(k+3)}{3}}_{\text  {Right side of }S(k+1)}


    We have established S(k+1) . . . The inductive proof is complete.

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