1. Tardigrade
  2. Question
  3. Mathematics
  4. Let Vr denotes the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is (2 r-1). Let Tr=Vr+1-Vr and Qr=Tr+1-Tr for r=1,2, ldots Tr is always (a) an odd number (b) an even number (c) a prime number (d) a composite number

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Q. Let $V_{r}$ denotes the sum of the first $r$ terms of an arithmetic progression (AP) whose first term is $r$ and the common difference is $(2 r-1)$. Let $T_{r}=V_{r+1}-V_{r}$ and $Q_{r}=T_{r+1}-T_{r}$ for $r=1,2, \ldots$
$T_{r}$ is always
(a) an odd number
(b) an even number
(c) a prime number
(d) a composite number

IIT JEEIIT JEE 2007Sequences and Series

Solution:

$ V_{r+1} - V_r = (r+1)^3 - r^3 - \frac{1}{2} [(r+1)^2 -r^2] +\frac{1}{2} $
$ = 3r^2 + 2r - 1$
$ \therefore T_r= 3r^2 + 2r - 1 = (r+1)\, (3r-1)$
which is a composite number