Question

Find the common difference of the Arithmetic progression $\frac{1}{a},\frac{3-a}{3a},\frac{3-2a}{3a},...\left(a\ne 0\right)$

Answer 1

Given: $\frac{1}{a},\frac{3-a}{3a},\frac{3-2a}{3a},...\left(a\ne 0\right)$


$$\begin{gathered} \mathrm{Common}\mathrm{difference}=a_2-a_1 \\ =\frac{3-a}{3a}-\frac{1}{a} \\ =\frac{3-a-3}{3a} \\ =\frac{-a}{3a} \\ =\frac{-1}{3} \end{gathered}$$

Hence, the common difference of the Arithmetic progression $\frac{1}{a},\frac{3-a}{3a},\frac{3-2a}{3a},...\left(a\ne 0\right)$ is $-\frac{1}{3}.$