(1)

\begin{align*} \sum_{k=1}^{n}\left\{ a_{1}+\left(k-1\right)d\right\} & =\left(a_{1}-d\right)\sum_{k=1}^{n}1+d\sum_{k=1}^{n}k\\ & =\left(a_{1}-d\right)n+\frac{dn\left(1+n\right)}{2}\\ & =\frac{1}{2}n\left\{ 2a_{1}+\left(n-1\right)d\right\} \end{align*}

(2)

\begin{align*} \sum_{k=1}^{n}\left(a_{1}r^{k-1}\right) & =a_{1}+\sum_{k=2}^{n}\left(a_{1}r^{k-1}\right)\\ & =a_{1}+\sum_{k=1}^{n-1}\left(a_{1}r^{k}\right)\\ & =a_{1}+r\sum_{k=1}^{n-1}\left(a_{1}r^{k-1}\right)\\ & =a_{1}-a_{1}r^{n}+r\LHS\\ & =a_{1}\frac{1-r^{n}}{1-r} \end{align*}

(3)

(2)より、
\begin{align*} \sum_{k=1}^{\infty}\left(a_{1}r^{k-1}\right) & =\lim_{n\rightarrow\infty}a_{1}\frac{1-r^{n}}{1-r}\\ & =\frac{a_{1}}{1-r}\cmt{\because\left|r\right|<1\leftrightarrow\lim_{n\rightarrow\infty}r^{n}=0} \end{align*}