Q. 32 - Calculus

Question

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$\left\{\frac{n}{2n+1}\right\}$

Step-by-Step Solution

Answer

The first five terms are $\frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{4}{9},\frac{5}{11}.$

Step 1. Given information.

The given series is $\left\{\frac{n}{2n+1}\right\}$.

Step 2. First term.

The general sequence is $a_n=\frac{n}{2n+1}$

Therefore, when $n=1,$

$$\begin{gathered} a_1=\frac{1}{2\left(1\right)+1} \\ =\frac{1}{3} \end{gathered}$$

Step 3. Remaining terms.

Now put,

$$\begin{gathered} n=2, \\ {a}_2=\frac{2}{2\left(2\right)+1}=\frac{2}{5} \\ n=3, \\ {a}_3=\frac{3}{2\left(3\right)+1}=\frac{3}{7} \\ n=4, \\ {a}_4=\frac{4}{2\left(4\right)+1}=\frac{4}{9} \\ n=5, \\ {a}_5=\frac{5}{2\left(5\right)+1}=\frac{5}{11} \end{gathered}$$