- #1

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## Homework Statement

Write an explicit formula for the sequence determined by the following recursion formula.

t[itex]_{1}[/itex]= 0; t[itex]_{n}[/itex] = t[itex]_{n-1}[/itex] + [itex]\frac{2}{n(n+1)}[/itex]

## The Attempt at a Solution

t[itex]_{1}[/itex] = 0

t[itex]_{2}[/itex] = t[itex]_{1}[/itex] + [itex]\frac{2}{2(2+1)}[/itex]

t[itex]_{2}[/itex] = [itex]\frac{1}{3}[/itex]

t[itex]_{3}[/itex] = t[itex]_{2}[/itex] + [itex]\frac{2}{3(3+1)}[/itex]

t[itex]_{3}[/itex] = [itex]\frac{1}{3}[/itex] + [itex]\frac{2}{3(3+1)}[/itex]

t[itex]_{3}[/itex] = [itex]\frac{4}{12}[/itex] + [itex]\frac{2}{12)}[/itex]

t[itex]_{3}[/itex] = [itex]\frac{1}{2}[/itex]

t[itex]_{4}[/itex] = t[itex]_{3}[/itex] + [itex]\frac{2}{4(4+1)}[/itex]

t[itex]_{4}[/itex] = [itex]\frac{1}{2}[/itex] + [itex]\frac{2}{20}[/itex]

t[itex]_{4}[/itex] = [itex]\frac{3}{5}[/itex]

My sequence is 0, [itex]\frac{1}{3}[/itex], [itex]\frac{1}{2}[/itex], [itex]\frac{3}{5}[/itex] [itex]\cdots[/itex]

How do I make an explicit formula if there is no common difference nor a common ratio?