Question

Let `T_r` be the rth term of an A.P. whose first term is -1/2 and common difference is 1, then `sum_(r=1)^n sqrt(1+ T_r T_(r+1) T_(r+2) T_(r+3))`

Answer 1

Here, ` a= -1/2 and d = 1`
`:. T_r = a+(r-1)d = -1/2+(r-1)1 = r-3/2`
`:. T_rT_(r+1)T_(r+2)T_(r+3) = (r-3/2)(r-1/2)(r+1/2)(r+3/2)`
`=> T_rT_(r+1)T_(r+2)T_(r+3) = (r-3/2)(r+3/2)(r-1/2)(r+1/2)`
`=> T_rT_(r+1)T_(r+2)T_(r+3) = (r^2-9/4)(r^2-1/4)`
`=> T_rT_(r+1)T_(r+2)T_(r+3) = r^4-10/4r^2+9/16`
`:. 1+T_rT_(r+1)T_(r+2)T_(r+3) = r^4-10/4r^2+25/16 = (r^2-5/4)^2`
`:. sum_(r=1)^n sqrt(1+T_rT_(r+1)T_(r+2)T_(r+3) )= sum_(r=1)^n (r^2-5/4)`
`= sum_(r=1)^n r^2 - sum_(r=1)^n 5/4`
`= (n(n+1)(2n+1))/6-5/4n`
So, option `A` is the correct answer.