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Question

Find the nth term of the sequence
$$\dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, ..........$$


Solution

In the given sequence is $$\cfrac { 1 }{ 2 } ,\cfrac { 2 }{ 3 } ,\cfrac { 3 }{ 4 } ,......$$, the first term $$t_{ 1 }=\cfrac { 1 }{ 2 }$$, second term $$t_{ 2 }=\cfrac { 2 }{ 3 }$$ and third term $$t_{ 3 }=\cfrac { 3 }{ 4 }$$

We observe that:

$$t_{ 1 }=\cfrac { 1 }{ 1+1 } =\cfrac { 1 }{ 2 }$$ and 
$$t_{ 2 }=\cfrac { 2 }{ 2+1 } =\cfrac { 2 }{ 3 }$$ and so on...

Therefore, the given sequence is of the form $$t_{ n }=\cfrac { n }{ n+1 }$$.
 
Hence, the nth term of the sequence is $$t_{ n }=\cfrac { n }{ n+1 }$$.

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