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

Question

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 (A) an odd number (B) an even number (C) a prime number (D) a composite number

Tardigrade Admin · Accepted Answer

Vr+1 - Vr = (r+1)3 - r3 - (1/2) [(r+1)2 -r2] +(1/2) = 3r2 + 2r - 1 ∴ Tr= 3r2 + 2r - 1 = (r+1) (3r-1) which is a composite number

an odd number

an even number

a prime number

a composite number

Vr+1​−Vr​=(r+1)3−r3−21​[(r+1)2−r2]+21​ =3r2+2r−1 ∴Tr​=3r2+2r−1=(r+1)(3r−1) which is a composite number

$V_{r + 1} - V_{r} = (r + 1)^{3} - r^{3} - \frac{1}{2} [(r + 1)^{2} - r^{2}] + \frac{1}{2}$
$= 3 r^{2} + 2 r - 1$
$∴ T_{r} = 3 r^{2} + 2 r - 1 = (r + 1) (3 r - 1)$
which is a composite number