$\begin{align}  & 1+r+{{r}^{2}}+{{r}^{3}}+---+{{r}^{n-1}}=127---(i) \\ & 1+\tfrac{1}{r}+\tfrac{1}{{{r}^{2}}}+\tfrac{1}{{{r}^{3}}}+---+\tfrac{1}{{{r}^{n-1}}}=\frac{127}{64}---(ii) \\ & \text{Sum of }n-\text{terms in }(i) \\ & \frac{1({{r}^{n}}-1)}{r-1}=127 \\ & {{r}^{n}}-1=127r-127 \\ & {{r}^{n}}=127r-126----(iii) \\ & \text{Sum of the }n-\text{terms in }(ii) \\ & \frac{1\left[ 1-{{(\tfrac{1}{r})}^{n}} \right]}{1-\tfrac{1}{r}}=\frac{127}{64} \\ & 64\left[ 1-\tfrac{1}{{{r}^{n}}} \right]=127(1-\tfrac{1}{r}) \\ & 64-64{{r}^{-n}}=127-127{{r}^{-1}} \\ & 127{{r}^{-1}}-64{{r}^{-n}}=63----(iv) \\ & \text{From }(iii)\text{ }{{r}^{n}}=127r-126\text{ and substituted in }(iv) \\ & 127{{r}^{-1}}-64{{r}^{-n}}=63 \\ & \frac{127}{r}-\frac{64}{{{r}^{n}}}=63 \\ & \frac{127}{r}-\frac{64}{127r-126}=63 \\ & 127(127r-126)-64r=63r(127r-126) \\ & -16002+16129r-64=7938r+8001{{r}^{2}} \\ & 8001{{r}^{2}}-24003r+16002=0 \\ & {{r}^{2}}-3r+20=0 \\ & (r+1)(r-2)=0 \\ & \text{By inspection }r=2 \\ & \text{when }r=2 \\ & 127=\frac{{{2}^{n}}-1}{2-1} \\ & 127={{2}^{n}}-1 \\ & 128={{2}^{n}} \\ & {{2}^{7}}={{2}^{n}} \\ & n=7 \\\end{align}$