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Math Help - Proof By Induction

  1. #1
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    Proof By Induction

    Can somone solve this and show how to do it.

    1/1x2 + 1/2x3 +...+ 1/n(n+1) = n/n+1
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  2. #2
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    Hello, parkerj3231!

    Prove by induction: . \frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \hdots + \frac{1}{n(n+1)} \;=\;\frac{n}{n+1}

    Verify S(1)\!:\;\;\frac{1}{1\cdot2} \:=\:\frac{1}{1+1} . . . true.


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


    Add \frac{1}{(k+1)(k+2)} to both sides:

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


    The right side is: . \frac{k}{k+1}\cdot{\color{blue}\frac{k+2}{k+2}} + \frac{1}{(k+1)(k+2)} \;\;=\;\;\frac{k(k+2) + 1}{(k+1)(k+2)} \;\;=\;\;\frac{k^2 + 2k + 1}{(k+1)(k+2)}

    . . . . . . . . . . = \;\;\frac{(k+1)^2}{(k+1)(k+2)} \;\;=\;\;\frac{k+1}{k+2} \quad\Leftarrow\;\text{ This is the right side of }S(k+1)


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

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