Question

Find a general term for the sequence, $-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11},...$

Short answer

The general form of the sequence $-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11},...$is $a_n=\frac{2n}{2n+1}$

Step-by-step solution

Step 1. Given

The sequence $-\frac{2}{3},-\frac{4}{5},-\frac{6}{7},-\frac{8}{9},-\frac{10}{11},...$

To find the general form of the sequence.

Step 2. Look for a pattern in the terms.

Let us look for a pattern in the terms.

$-\frac{2\left(1\right)}{2\left(1\right)+1},-\frac{2\left(2\right)}{2\left(2\right)+1},-\frac{2\left(3\right)}{2\left(3\right)+1},-\frac{2\left(4\right)}{2\left(4\right)+1},-\frac{2\left(5\right)}{2\left(5\right)+1},...$

Step 3. General form of the sequence

By observing the pattern, we can find that he numerator is twice the number of terms and the denominator is one more than the twice the number of terms.

So, the general form is $a_n=\frac{2n}{2n+1}$