می‌دانيم جملهٔ $n$‌ام یک دنباله‌ی حسابی از رابطه‌ی ${{t}_{n}}={{t}_{1}}+(n-1)d$ بدست می‌آید، پس:

$\left\{ \begin{matrix}
{{t}_{12}}+{{t}_{10}}=5  \\
{{t}_{12}}+{{t}_{10}}=25  \\
\end{matrix} \right.\,\,\Rightarrow \left\{ \begin{matrix}
({{t}_{1}}+11d)-({{t}_{1}}+9d)=5  \\
({{t}_{1}}+11d)-({{t}_{1}}+9d)=25  \\
\end{matrix} \right.$

بنابراین:

$\Rightarrow \left\{ \begin{matrix}
2d=5\Rightarrow d=2/5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \\
2{{t}_{1}}+20d=25\xrightarrow{d=2/5}2{{t}_{1}}+20\times (2/5)=25  \\
\Rightarrow {{t}_{1}}=-12/5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  \\
\end{matrix} \right.$

پس جملهٔ بيست و يكم برابر است با:

${{t}_{21}}={{t}_{1}}+20d=-12/5+20\times (2/5)=37/5$