Let Vr denote the sum of the first ' r ' terms of an arithmetic progression (A.P.) whose first term is ' r ' and the common difference is (2 r -1). Let T r = V r +1- V r -2 and Q r = T r +1- T r for r =1,2, ldots Tr is always

Question

Let Vr denote the sum of the first ' r ' terms of an arithmetic progression (A.P.) whose first term is ' r ' and the common difference is (2 r -1). Let T r = V r +1- V r -2 and Q r = T r +1- T r for r =1,2, ldots Tr is always (A) an odd number (B) an even number (C) a prime number (D) a composite number

Tardigrade Admin · Accepted Answer

V r =( r /2)(2 r 2- r +1)= r 3-(1/2) r 2+( r /2) T r =(( r +1)3- r 3)-(1/2)(( r +1)2- r 2)+(1/2)( r +1- r )-2 =( r 2+2 r +1+ r 2+ r 2+ r )-(1/2)(2 r +1)+(1/2)-2 T r =3 r 2+3 r +1- r -2=3 r 2+2 r -1 T r =3 r 2+3 r - r -1=(3 r -1)( r +1) ⇒ text (D)

Q. Let Vr​ 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 Tr​=Vr+1​−Vr​−2 and Qr​=Tr+1​−Tr​ for r=1,2,… Tr​ is always

an odd number

an even number

a prime number

a composite number

Vr​=2r​(2r2−r+1)=r3−21​r2+2r​ Tr​=((r+1)3−r3)−21​((r+1)2−r2)+21​(r+1−r)−2 =(r2+2r+1+r2+r2+r)−21​(2r+1)+21​−2 Tr​=3r2+3r+1−r−2=3r2+2r−1 Tr​=3r2+3r−r−1=(3r−1)(r+1)⇒ (D)

Q. 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 $(2 r - 1)$ .
Let $T_{r} = V_{r + 1} - V_{r} - 2$ and $Q_{r} = T_{r + 1} - T_{r}$ for $r = 1, 2, \dots$
$T_{r}$ is always