Writing the Terms of a Sequence

Write the first three terms of each sequence:

1. \(\{b_{n}\} _{n=1}^{\infty }=\left\{ \dfrac{1}{3n-2}\right\} _{n=1}^{\infty }\)
2. \(\left\{ c_{n}\right\}=\left\{ \dfrac{2n-1}{n^{3}}\right\}\)

Solution (a) The \(n\)th term of this sequence is \(b_{n}=\dfrac{1}{3n-2}.\) The first three terms are \[ b_{1}=\dfrac{1}{3\cdot 1-2}=1\qquad b_{2}=\dfrac{1}{3\cdot 2-2}=\dfrac{1}{4}\qquad b_{3}=\dfrac{1}{3\cdot 3-2}=\dfrac{1}{7} \]

(b) The \(n\)th term of this sequence is \(c_{n}=\dfrac{2n-1}{n^{3}}\). Then \[ c_{1}=\dfrac{2(1)-1}{1^{3}}=1\qquad c_{2}=\dfrac{2(2) -1}{2^{3}}=\dfrac{3}{8}\qquad c_{3}=\dfrac{2 (3) -1}{3^{3}}=\dfrac{5}{27} \]