میدانیم ${{t}_{n}}={{t}_{1}}{{r}^{n-1}}$، با توجه به فرض مسئله داریم:
$\frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{2}}}+\frac{1}{{{t}_{3}}}=\frac{21}{48}\Rightarrow \frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{1}}r}+\frac{1}{{{t}_{1}}{{r}^{2}}}=\frac{21}{48}\Rightarrow \frac{{{r}^{2}}+r+1}{{{t}_{1}}{{r}^{2}}}=\frac{21}{48}\xrightarrow{(*)}\frac{\frac{63}{{{t}_{1}}}}{{{t}_{1}}{{r}^{2}}}=\frac{21}{48}\Rightarrow \frac{63}{{{t}_{1}}^{2}{{r}^{2}}}=\frac{21}{48}\Rightarrow {{t}_{1}}^{2}{{r}^{2}}=\frac{63\times 48}{21}\Rightarrow {{t}_{1}}^{2}{{r}^{2}}=144\Rightarrow {{t}_{1}}r=\pm 12$