(1)

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

\(a_{c}\)が既知のとき

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

(2)

\(a_{1}\ne0\land r\ne0\)ならば\(a_{n}\ne0\)なので、
\begin{align*} a_{n} & =a_{1}\prod_{k=1}^{n-1}\frac{a_{k+1}}{a_{k}}\\ & =a_{1}\prod_{k=1}^{n-1}r\\ & =a_{1}r^{n-1} \end{align*}
\(a_{1}=0\lor r=0\)のときもこの式は成り立つ。ただし\(0^{0}=1\)とする。

\(a_{c}\)が既知のとき

\begin{align*} a_{n} & =a_{c}\prod_{k=c}^{n-1}\frac{a_{k+1}}{a_{k}}\\ & =a_{c}\prod_{k=c}^{n-1}r\\ & =a_{c}r^{n-c} \end{align*}

(2)別解

\(a_{1}\ne0\land r\ne0\)ならば\(a_{n}\ne0\)なので、
\begin{align*} a_{n} & =\exp\left(\log a_{n}\right)\\ & =\exp\left(\log a_{1}+\sum_{k=1}^{n-1}\left(\log a_{n+1}-\log a_{n}\right)\right)\\ & =\exp\left(\log a_{1}+\sum_{k=1}^{n-1}\log r\right)\\ & =\exp\left(\log\left(a_{1}r^{n-1}\right)\right)\\ & =a_{1}r^{n-1} \end{align*}
\(a_{1}=0\lor r=0\)のときもこの式は成り立つ。ただし\(0^{0}=1\)とする。

(3)

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

\(a_{c}\)が既知のとき

\begin{align*} a_{n} & =a_{c}+\sum_{k=c}^{n-1}\left(a_{k+1}-a_{k}\right)\\ & =a_{c}+\sum_{k=c}^{n-1}f\left(k\right) \end{align*}

(4)

\(a_{1}\ne0\land\prod_{k=1}^{n-1}f_{k}\ne0\)ならば\(a_{n}\ne0\)なので、
\begin{align*} a_{n} & =a_{1}\prod_{k=1}^{n-1}\frac{a_{k+1}}{a_{k}}\\ & =a_{1}\prod_{k=1}^{n-1}f\left(k\right) \end{align*}
\(a_{1}=0\lor\prod_{k=1}^{n-1}f_{k}=0\)のときもこの式は成り立つ。

\(a_{c}\)が既知のとき

\begin{align*} a_{n} & =a_{c}\prod_{k=c}^{n-1}\frac{a_{k+1}}{a_{k}}\\ & =a_{c}\prod_{k=c}^{n-1}f\left(k\right) \end{align*}

(4)別解

\(a_{1}\ne0\land\prod_{k=1}^{n-1}f_{k}\ne0\)ならば\(a_{n}\ne0\)なので、
\begin{align*} a_{n} & =\exp\left(\log a_{n}\right)\\ & =\exp\left(\log a_{1}+\sum_{k=1}^{n-1}\left(\log a_{k+1}-\log a_{k}\right)\right)\\ & =\exp\left(\log a_{1}+\sum_{k=1}^{n-1}\left(\log f\left(k\right)\right)\right)\\ & =\exp\left(\log\left(a_{1}\prod_{k=1}^{n-1}f\left(k\right)\right)\right)\\ & =a_{1}\prod_{k=1}^{n-1}f\left(k\right) \end{align*}
\(a_{1}=0\lor\prod_{k=1}^{n-1}f_{k}=0\)のときもこの式は成り立つ。