JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Sequences and Series

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

Which one of the following is a correct statement?

Answer: (B)

Solution

From the previous result,

T_{r} = 3 r^{2} + 2 r - 1

T_{r + 1} = 3 (r + 1)^{2} + 2 (r + 1) - 1 = 3 r^{2} + 6 r + 3 + 2 r + 2 - 1 = 3 r^{2} + 8 r + 4

Q_{r} = T_{r + 1} - T_{r} = (3 r^{2} + 8 r + 4) - (3 r^{2} + 2 r - 1) = 6 r + 5

Q_{r + 1} = 6 (r + 1) + 5 = 6 r + 11

Common difference:

Q_{r + 1} - Q_{r} = 6

Therefore,

Q_{1}, Q_{2}, Q_{3}, \dots

are in A.P. with common difference 6.

JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Single correct option