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Question Number 82995 by M±th+et£s last updated on 26/Feb/20

Commented by mr W last updated on 26/Feb/20

$$a_{1000}=\frac{1}{499501}?$$

Answered by mind is power last updated on 26/Feb/20

$$\Rightarrow \frac{1}{a_{n+1}}=n+\frac{1}{a_n}$$$$let{b}_n=\frac{1}{a_n}$$$$\Rightarrow b_{n+1}=n+b_n$$$$b_0=1$$$$\Rightarrow \underset{k=0}{\sum ^{n-1}}(b_{k+1}-b_k)=\underset{k=0}{\sum ^{n-1}}k$$$$\Rightarrow b_n-b_0=\frac{n}{2}(n-1)$$$$\Rightarrow b_n=1+\frac{n}{2}(n-1)$$$$\Rightarrow a_n=\frac{1}{b_n}=\frac{2}{n^2-n+2}$$$$a_{1000}=\frac{2}{999002}=\frac{1}{499501}$$$$$$

Commented by mr W last updated on 26/Feb/20

$$finesir!igotthesame.$$

Commented by M±th+et£s last updated on 26/Feb/20

$$thankyousir.thisisthevalue$$

Commented by mind is power last updated on 27/Feb/20

$$withepleasur$$

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