JEE Advanced2007MathematicsSequences and SeriesActual
Paragraph: Let V_r denotes 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, Question: The sum V₁+V₂+ +V_n is
Options
- A1 12 n(n+1) (3 n^2-n+1 )
- B1 12 n(n+1) (3 n^2+n+2 )
- C1 2 n (2 n^2-n+1 )
- D1 3 (2 n^3-2 n+3 )
Correct answer
B. 1 12 n(n+1) (3 n^2+n+2 )
Step-by-step solution
aligned & V_r= r 2 [2 r+(r-1)(2 r-1)]= 1 2 (2 r^3-r^2+r ) & V_r= 1 2 [2 r^3- r^2+ r ] & = 1 2 [2 ( n(n+1) 2 )^2- n(n+1)(2 n+1) 6 + n(n+1) 2 ] & = 1 12 n(n+1) (3 n^2+n+2 ) . & aligned