How to find the sum of $n$ terms in the sequence - $ \dfrac{1}{1.2.3}+ \dfrac{1}{2.3.4}+\dfrac{1}{3.4.5} + \cdots+ \dfrac{1}{n(n+1)(n+2)}$?

Asked Jun 19 '20 at 16:19

Active Jun 19 '20 at 16:25

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How to find the sum of $n$ terms in the sequence - $ \dfrac{1}{1.2.3}+$ $\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5} + \cdots+ \dfrac{1}{n(n+1)(n+2)}$?

I added a few first terms like, $ \dfrac{1}{6}, \dfrac{5}{24}, \dfrac{9}{40},\dfrac{7}{30}$.I just couldn't figure out any patterns.

Is there any generalized formula to get the sum of $n$ terms or what kind of approach should I go for to solve problems like this?

edited Jun 19 '20 at 16:25

asked Jun 19 '20 at 16:19

Shromi

3

Do you know about partial fraction decomposition? – user170231 Jun 19 '20 at 16:22
1

Please use \cdot to get a centered dot for multiplication, so 1 \cdot 2 \cdot 3 gives $1 \cdot 2 \cdot 3$. Periods can look like decimal points. – Ross Millikan Jun 19 '20 at 16:23

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