Answer
(a) the $n$th term is ${a_n} = \frac{{{{\left( { - 1} \right)}^{n + 1}}}}{{{n^3}}}$ for $n=1,2,3,....$
(b) the $n$th term is ${a_n} = \frac{{n + 1}}{{n + 5}}$ for $n=1,2,3,....$
Work Step by Step
(a) The sign of the numerator is alternating. If $n$ starts from 1, the general term of the numerator is ${\left( { - 1} \right)^{n + 1}}$. The general term of the denominator is $n^3$. Thus, the $n$th term is ${a_n} = \frac{{{{\left( { - 1} \right)}^{n + 1}}}}{{{n^3}}}$ for $n=1,2,3,....$
(b) If $n$ starts from 1, the numerator is always one more than the index, whereas the denominator is five more than the index. So, the $n$th term is ${a_n} = \frac{{n + 1}}{{n + 5}}$ for $n=1,2,3,....$
