Question

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

Answer 1

We can write the expression for the `n^(th)` term of the sequence as `a_n=(-1)^n(n+1)/(n+2)`

Explanation:

Let us divide this in three parts.

First - Numerator goes like `{2,3,4,5,6,..}`. Here, we have arithmetic sequence with first term is `2` and common difference as `1` and hence `a_n=2+(n-1)xx1=n+1`

Second - Denominator goes like `{3,4,5,6,7,..}`. Here, we have arithmetic sequence with first term is `3` and common difference as `1` and hence `a_n=3+(n-1)xx1=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 `(-1)^n`

Hence, we can write the expression for the `n^(th)` term of the sequence as

`a_n=(-1)^n(n+1)/(n+2)`