Answer
(a) the general term is $\frac{n}{{n + 1}}$ for $n = 1,2,3,...$
(b) the general term is $\cos \pi n$ for $n=1,2,3,...$
(c) the general term is ${\left( { - 1} \right)^{n + 1}}$ for $n=1,2,3,...$
(d) the general term is $\frac{{n!}}{{{2^n}}}$
Work Step by Step
(a) $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...$
Since the denominator is one more than the numerator, the general term is $\frac{n}{{n + 1}}$ for $n = 1,2,3,...$
(b) $-1,1,-1,1,...$
The sign is alternating. If it starts from n=1 then the general term is $\cos \pi n$ for $n=1,2,3,...$
(c) $1,-1,1,-1,...$
The sign is alternating. If it starts from $n=1$ the terms ${a_n}$ is 1 for odd n and -1 for even n.
So, the general term is ${\left( { - 1} \right)^{n + 1}}$ for $n=1,2,3,...$
(d) $\frac{1}{2},\frac{2}{4},\frac{6}{8},\frac{{24}}{{16}},...$
If it starts from $n=1$ the numerator is $n!$, whereas the denominator is $2^n$. Thus, the general term is $\frac{{n!}}{{{2^n}}}$.
