Question

In the following exercise, write the first five terms of the sequence whose general term is given.

$a_n=\frac{3^n}{n^3}$

Short answer

The first five terms of the sequence whose general term is $a_n=\frac{3^n}{n^3}$are$3,\frac{9}{8},1,\frac{81}{64},\frac{243}{125}.$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegeneraltermforthesequenceisgivenas}\\ a_n=\frac{3^n}{n^3}\\ \text{Wehavetofindthefirstfivetermsofthesequence.}\end{array}$

Step 2. Calculating the terms

$\begin{array}{l}\text{Wewillsubstitute}n=1,2,3,4\text{and}5\text{intheordertoobtainthefirstfivetermsofthesequence.}\\ \text{Now,}\end{array}$

$\begin{array}{cccccc}n:& 1& 2& 3& 4& 5\\ a_n:& \begin{array}{l}{a}_1=\frac{3^1}{1^3}\\ =\frac{3}{1}\\ =3\end{array}& \begin{array}{l}{a}_2=\frac{3^2}{2^3}\\ =\frac{9}{8}\end{array}& \begin{array}{l}{a}_3=\frac{3^3}{3^3}\\ =\frac{27}{27}\\ =1\end{array}& \begin{array}{l}{a}_4=\frac{3^4}{4^3}\\ =\frac{81}{64}\end{array}& \begin{array}{l}{a}_5=\frac{3^5}{5^3}\\ =\frac{243}{125}\end{array}\end{array}$

Step 3. Conclusion

From the above step, we can conclude that the first five terms of the sequence will be$3,\frac{9}{8},1,\frac{81}{64},\frac{243}{125}.$