Question

Find five rational numbers between $2$ and $3$ by mean method.

Answer 1

Compute the required rational numbers by mean method :

The rational number between $a$and $b$ can be obtained by calculating the mean of $a$ and $b$ i.e. $mean\left(a,b\right)=\frac{\left(a+b\right)}{2}$.

1. Rational number between $2$ and $3$ is $mean\left(2,3\right)=\frac{\left(2+3\right)}{2}=\frac{5}{2}$
2. Rational number between $2$ and $\frac{5}{2}$ is $mean\left(2,\frac{5}{2}\right)=\frac{\left(2+{\displaystyle \frac{5}{2}}\right)}{2}=\frac{9}{4}$
3. Rational number between $2$ and $\frac{9}{4}$ is $mean\left(2,\frac{9}{4}\right)=\frac{\left(2+{\displaystyle \frac{9}{4}}\right)}{2}=\frac{17}{8}$
4. Rational number between $\frac{5}{2}$ and $3$ is $mean\left(\frac{5}{2},3\right)=\frac{\left({\displaystyle \frac{5}{2}}+3\right)}{2}=\frac{11}{4}$
5. Rational number between $\frac{11}{4}$ and $3$ is $mean\left(\frac{11}{4},3\right)=\frac{\left({\displaystyle \frac{11}{4}}+3\right)}{2}=\frac{23}{8}$

Hence, the five rational numbers between $2$ and $3$ are $\frac{5}{2},\frac{9}{4},\frac{17}{8},\frac{11}{4}$ and $\frac{23}{8}$.