Question

write the set x={5,13,25,41} in set builder form​

Answer 1

$\large \text{X \ =\{5,\ 13,\ 25,\ 41\} }$

First we have to write each element in the set X in relation with natural numbers.

It is found that,

$\displaystyle \begin{aligned}5&=\ \ 1+4&=&\ \ 1+4(1)&=&\ \ 1+4\sum_{n=1}^{1}n\\ \\ 13&=\ \ 1+4+8&=&\ \ 1+4(1+2)&=&\ \ 1+4\sum_{n=1}^{2}n\\ \\ 25&=\ \ 1+4+8+12&=&\ \ 1+4(1+2+3)&=&\ \ 1+4\sum_{n=1}^{3}n\\ \\ 41&=\ \ 1+4+8+12+16&=&\ \ 1+4(1+2+3+4)&=&\ \ 1+4\sum_{n=1}^{4}n\end{aligned}$

Hence the nth element of the set is in the form of,

$\displaystyle 1+4\sum_{k=1}^{n}k\ \ \ \ \ \Longrightarrow\ \ \ \ \ 1+4\cdot \frac{n(n+1)}{2}\ \ \ \ \ \Longrightarrow\ \ \ \ \ 2n(n+1)+1$

Here it seems that the least and highest values of n are 1 and 4 respectively. And also, n is a natural number.

Hence the set X in set builder form will be,

$\text{X \ =\{x:x=2n(n+1)+1,\ n is a natural number and 1\leq n\leq 4 \}}$

OR

$\text{X \ = \{x:x=2n(n+1)+1,\ n\in \mathbb{N},\ 1\leq n\leq 4\} }$