Question

Which of the following are $A{P}^{\prime }s$? If they form an $AP$, find the common difference $d$ and the first $3$ terms.
$3,3+$$\sqrt{2}$, $3+$ 2$\sqrt{2}$, $3+$3$\sqrt{2}$,...

• A. It is not an $A.P.$
  
• B. It is an $A.P$ and $d=\sqrt{2}$, $t_5=3+4\sqrt{2}$, $t_6=3+5\sqrt{2}$, $t_7=3+6\sqrt{2}$
  
• C. It is an $A.P$ and $d=\sqrt{3}$, $t_5=3+4\sqrt{2}$, $t_6=4+5\sqrt{2}$, $t_7$= 5 + 6\sqrt {2}$
• D. None of these
  

Answer 1

The correct option is B It is an $A.P$ and $d=\sqrt{2}$, $t_5=3+4\sqrt{2}$, $t_6=3+5\sqrt{2}$, $t_7=3+6\sqrt{2}$

Given series is $3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2},.......$

Clearly, the given sequence is an AP with first term $a=3$ and common difference $d=3+\sqrt{2}-3=\sqrt{2}$
$n^{th}term=a_n=a+(n-1)d$
$\therefore a_5=a+4d=3+4\times \sqrt{2}=3+4\sqrt{2}$
$a_6=a+5d=3+5\times \sqrt{2}=3+5\sqrt{2}$
$a_7=a+6d=3+6\times \sqrt{2}=3+6\sqrt{2}$