Question

Question 4 (xii)
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(xii) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32}\dots \dots$

Answer 1

(xii) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32}\dots \dots$
Here,
$a_2-a_1=\sqrt{8}-\sqrt{2}=2\sqrt{2}-\sqrt{2}=\sqrt{2}{a}_3-a_2=\sqrt{18}-\sqrt{8}=3\sqrt{2}-2\sqrt{2}=\sqrt{2}{a}_4-a_3=\sqrt{32}-\sqrt{18}=4\sqrt{2}-3\sqrt{2}=\sqrt{2}$
$\Rightarrow a_{n+1}-a_n$ is same everytime.
Therefore,$d=\sqrt{2}$ and the given numbers are in A.P.
Next three terms are:
$a_5=\sqrt{32}+\sqrt{2}=4\sqrt{2}+\sqrt{2}=5\sqrt{2}=\sqrt{50}{a}_6=5\sqrt{2}+\sqrt{2}=6\sqrt{2}=\sqrt{72}{a}_7=6\sqrt{2}+\sqrt{2}=7\sqrt{2}=\sqrt{98}$