IIT-JEE 2007 · Paper 1 · Mathematics Q59

← IIT-JEE 2007 · Paper 1

MathematicsSingle correct+4 / -1Question 59 of 66

Passage

Paragraph for Question Nos. 58 to 60

Let $V_r$ denote the sum of the first $r$ terms of an arithmetic progression (A.P.) whose first term is $r$ and the common difference is $(2r-1)$ . Let
$T_r=V_{r+1}-V_r-2\quad\text{and}\quad Q_r=T_{r+1}-T_r\quad\text{for }r=1,2,\ldots$

$T_r$ is always

an odd number

an even number

a prime number

a composite number

Answer and worked solution

Official answer: D

Difference. From $V_r=\frac{2r^3-r^2+r}2$ ,
$V_{r+1}-V_r=\frac{2(3r^2+3r+1)-(2r+1)+1}2=3r^2+2r+1.$

So $T_r=3r^2+2r-1=(3r-1)(r+1)$ . Both factors are at least 2 for $r\ge1$ , so $T_r$ is always composite ( $T_1=4$ , $T_2=15$ , …). It is not always odd or always even.

(D)

Topics:Arithmetic progression

← Previous question Next question →

IIT-JEE 2007 · Paper 1, Mathematics question 59