Answer
$1,\ \displaystyle \frac{3}{2},\ \displaystyle \frac{7}{4},\ \displaystyle \frac{15}{8},\ \displaystyle \frac{31}{16},\ \displaystyle \frac{63}{32},\ \displaystyle \frac{127}{64},\ \displaystyle \frac{255}{128},\ \displaystyle \frac{511}{256},\ \displaystyle \frac{1023}{512}$
Work Step by Step
$\left[\begin{array}{llll}
n & a_{n+1} & =a_{n}+1/2^{n} & a_{n}\\ \\
1 & & & 1\\ \\
2 & a_{1}+\frac{1}{2^{1}} & 1+\frac{1}{2}= & \frac{3}{2} \\ \\
3 & a_{2}+\frac{1}{2^{2}}= & \frac{3}{2}+\frac{1}{4}= & \frac{7}{4}\\ \\
4 & a_{3}+\frac{1}{2^{3}}= & \frac{7}{4}+\frac{1}{8}= & \frac{15}{8}\\ \\
5 & a_{4}+\frac{1}{2^{4}}= & \frac{15}{8}+\frac{1}{16}= & \frac{31}{16}\\ \\
6 & a_{5}+\frac{1}{2^{5}}= & \frac{31}{16}+\frac{1}{32}= & \frac{63}{32}\\ \\
7 & a_{6}+\frac{1}{2^{6}}= & \frac{63}{32}+\frac{1}{64}= & \frac{127}{64}\ \\
8 & a_{7}+\frac{1}{2^{7}}= & \frac{127}{64}+\frac{1}{128}= & \frac{255}{128}\\ \\
9 & a_{8}+\frac{1}{2^{8}}= & \frac{255}{128}+\frac{1}{256}= & \frac{511}{256}\\ \\
10 & a_{9}+\frac{1}{2^{9}}= & \frac{511}{256}+\frac{1}{512}= & \frac{1023}{512}
\end{array}\right]$
