# How do you write the expression for the nth term of the sequence given -2/3, 3/4, -4/5, 5/6, -6/7,...?

Mar 6, 2017

We can write the expression for the ${n}^{t h}$ term of the sequence as ${a}_{n} = {\left(- 1\right)}^{n} \frac{n + 1}{n + 2}$

#### Explanation:

Let us divide this in three parts.

First - Numerator goes like $\left\{2 , 3 , 4 , 5 , 6 , . .\right\}$. Here, we have arithmetic sequence with first term is $2$ and common difference as $1$ and hence ${a}_{n} = 2 + \left(n - 1\right) \times 1 = n + 1$

Second - Denominator goes like $\left\{3 , 4 , 5 , 6 , 7 , . .\right\}$. Here, we have arithmetic sequence with first term is $3$ and common difference as $1$ and hence ${a}_{n} = 3 + \left(n - 1\right) \times 1 = n + 2$

Third - The sign of every odd term is minus and every even term is plus and we can say that this is given by ${\left(- 1\right)}^{n}$

Hence, we can write the expression for the ${n}^{t h}$ term of the sequence as

${a}_{n} = {\left(- 1\right)}^{n} \frac{n + 1}{n + 2}$