Prove that 1/(1*2) + 1/(2*3) + 1/(3*4) +...+1/(n(n+1)) = (n/n+1), for all n is an element of N. 

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lfryerda | High School Teacher | (Level 2) Educator

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We can prove that this is true using induction on `n in N` .  Starting with n=1, we see that the left side is:

`LS=1/{1 cdot 2}=1/2`

`RS=1/{1+1}=1/2=LS`

So the statement is true for n=1.

Now assume it is true for `n=k` .  This means that we have:

`1/{1(2)}+1/{2(3)}+1/{3(4)}+cdots+1/{k(k+1)}=k/{k+1}`

Now we need to prove that the statement is true for `n=k+1` .

Starting with the LS, we need to prove that it is equal to the `RS={k+1}/{k+2}` .

`LS=1/{1(2)}+1/{2(3)}+1/{3(4)}+cdots+1/{k(k+1)}+1/{(k+1)(k+2)}`   use assumption

`=k/{k+1}+1/{(k+1)(k+2)}`   simplify

`={k(k+2)+1}/{(k+1)(k+2)}`   simplify numerator

`={k^2+2k+1}/{(k+1)(k+2)}`   factor numerator

`={(k+1)^2}/{(k+1)(k+2)}`   cancel common factors

`={k+1}/{k+2}`

`=RS`

The statement is has been proved as true by induction.

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