Question

Question 4 (v)
Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(v) $3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2}$

Answer 1

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