Question

In Problems 29–36, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$suggested by the pattern.

role="math" localid="1646738671855" $\frac{1}{1·2},\frac{1}{2·3},\frac{1}{3·4},\frac{1}{4·5},...$

Short answer

The nth term of the sequence is $a_n=\frac{1}{n(n+1)}$

Step-by-step solution

Step 1. Write the given information.

The given sequence is$\frac{1}{1·2},\frac{1}{2·3},\frac{1}{3·4},\frac{1}{4·5},...$

Step 2. Determine the pattern of the sequence and write down the nth term. 

$a_1=\frac{1}{1·2},a_2=\frac{1}{2·3},a_3=\frac{1}{3·4}$

The pattern of the sequence is:

$$\begin{gathered} a_1=\frac{1}{1·2}=\frac{1}{1(1+1)} \\ {a}_2=\frac{1}{2·3}=\frac{1}{2(2+1)} \\ {a}_3=\frac{1}{3·4}=\frac{1}{3(3+1)} \\ {a}_n=\frac{1}{n(n+1)} \end{gathered}$$