For a ∈ R, |a| > 1, let limn→∞ Σr=1n √r / n7/3 ( Σr=1n 1/(na+r)2 ) = 54. Then possible value(s) a is/are

Correct option is B. -9
limn→∞ Σr=1n (r/n)1/2 / n7/3 ( Σr=1n 1/(na+r)2 ) = 54
∫01 x1/2 dx ∫01 dx/(a+x)2 = [ (3/4)x4/3 ]01 [ -1/(a+x) ]01 = (3/4) ( -1/(a+1) + 1/a ) = 54
(3/4) (1/a - 1/(a+1)) = 54
1/a(a+1) = (4/3) * 54
1/a(a+1) = 72
a(a+1) = 1/72
This equation is not correct. Let's re-examine the solution.
limn→∞ (1/n7/3) Σr=1n √r Σr=1n 1/(na+r)2 = 54
≈ (1/n7/3) ∫0n √x dx ∫0n 1/(na+x)2 dx
≈ (1/n7/3) [ (2/3)x3/2 ]0n [ -1/(na+x) ]0n
≈ (1/n7/3) (2/3)n3/2 (-1/(na+n) + 1/na)
≈ (2/3) (1/n7/3) n3/2 (n/(na(na+n)))
≈ (2/3) (1/n7/3) n5/2 (1/(n2a(a+1)))
≈ (2/3) (1/n7/3) n5/2 (1/(n2a(a+1)))
≈ (2/(3a(a+1)))
(2/(3a(a+1))) = 54
1/(a(a+1)) = 81
a(a+1) = 1/81
a2 + a - 1/81 = 0
Solving the quadratic equation, we get a ≈ -1 or a ≈ 0.0123.
However, neither of these are in the options, and the condition |a|>1 must be satisfied. Let's check the calculation again.
∫01 x1/2 dx ∫01 dx/(a+x)2 = (2/3) [-1/(a+x)]01 = (2/3)(1/a - 1/(a+1)) = (2/3)(1/a(a+1)) = 54
1/a(a+1) = 81
a(a+1) = 1/81
Solving this quadratic equation, we get a = -1 ± √(1 + 4/81)/2 which approximately equals -1, or 0.0123. Neither fits the condition |a|>1 and is among the given options.