Question

Which of the following is a correct statement:

• A. $Q_1,Q_2,Q_3,..$ are in A.P with common difference $5$
• B. $Q_1,Q_2,Q_3,..$ are in A.P with common difference $6$
• C. $Q_1,Q_2,Q_3,..$ are in A.P with common difference $11$
• D. $Q_1=Q_2=Q_3=...$

Answer 1

Answer: (D)

$Q_1=Q_2=Q_3=...$

$Q_r=T_{r+1}-T_r$
$\Rightarrow T_{r+1}-T_r=\left(3\right(r+1)-1)\left(\right(r+1)+1)-(3r-1)(r+1)$
On simplification we get
$T_{r+1}-T_r=(3r+2)(r+2)-(3r-1)(r+1)$
$=3r^2+6r+2r+4-(3r^2+3r-r-1)$
$=6r+5$ (on evaluation)and $d=Q_r-Q_{r-1}$
$=6r+5-\left(6\right(r-1)+5)$
$=6r+5-6r+6-5$ .... (on evaluation)
$=6$ ....... (on simplification)