Maths Question:
$\begin{align} & \text{If the tenth term of a G}\text{.P is 2 and the twentieth term is }\frac{1}{512},\text{ find the first term, } \\ & \text{common ratio and sum to infinity}\text{.} \\\end{align}$
Maths Solution:
$\begin{align} & {{U}_{n}}=a{{r}^{n-1}} \\ & {{U}_{10}}=a{{r}^{9}}=2 \\ & {{U}_{20}}=a{{r}^{19}}=\frac{1}{512} \\ & \frac{{{U}_{20}}}{{{U}_{10}}}=\frac{a{{r}^{19}}}{a{{r}^{9}}}=\frac{1}{1024} \\ & {{r}^{10}}=\frac{1}{{{2}^{10}}} \\ & r=\frac{1}{2} \\ & \text{Substitute }r=\frac{1}{2}\text{ into equation }(i) \\ & a{{\left( \frac{1}{2} \right)}^{9}}=2 \\ & a=2\times {{2}^{9}}={{2}^{10}}=1024 \\ & \text{Sum to Infinity}={{S}_{\infty }}=\frac{a}{1-r} \\ & {{S}_{\infty }}=\frac{1024}{1-\tfrac{1}{2}}=\frac{1024}{\tfrac{1}{2}}=2048 \\\end{align}$
University mathstopic: