Write the Set in Set-Builder Form
Write the set
\[ \left\{ \frac12, \frac25, \frac3{10}, \frac4{17}, \frac5{26}, \frac6{37}, \frac7{50} \right\} \]
in set-builder form.
Solution
Observe the pattern:
\[ \frac12=\frac1{1^2+1} \]
\[ \frac25=\frac2{2^2+1} \]
\[ \frac3{10}=\frac3{3^2+1} \]
Therefore, the general term is
\[ \frac{n}{n^2+1} \]
Hence, the set-builder form is
\[ \left\{ x:x=\frac{n}{n^2+1}, \ 1\le n\le7, \ n\in N \right\} \]