Question

Write the next four terms of the following A.P. $\frac{1}{6},\frac{1}{3},\frac{1}{2}$,

Answer 1

In the given progression is $\frac{1}{6},\frac{1}{3},\frac{1}{2},......$, the first term $a_1=\frac{1}{6}$, second term $a_2=\frac{1}{3}$ and third term $a_3=\frac{1}{2}$. So, the difference between the two terms of the given progression is:

$d=a_2-a_1=\frac{1}{3}-\frac{1}{6}=\frac{(1\times 2)-(1\times 1)}{3\times 2}=\frac{2-1}{6}=\frac{1}{6}$

Therefore, the next four terms of the arithmetic progression are:

$a_4=a_3+\frac{1}{6}=\frac{1}{2}+\frac{1}{6}=\frac{(1\times 3)+(1\times 1)}{2\times 3}=\frac{3+1}{6}=\frac{4}{6}=\frac{2}{3}$

$a_5=a_4+\frac{1}{6}=\frac{2}{3}+\frac{1}{6}=\frac{(2\times 2)+(1\times 1)}{3\times 2}=\frac{4+1}{6}=\frac{5}{6}$

$a_6=a_5+\frac{1}{6}=\frac{5}{6}+\frac{1}{6}=\frac{5+1}{6}=\frac{6}{6}=1$

$a_7=a_6+\frac{1}{6}=1+\frac{1}{6}=\frac{6+1}{6}=\frac{7}{6}$

Hence, the next four terms are $\frac{2}{3},\frac{5}{6},1,\frac{7}{6}$.