Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>lim
n→∞
(12 −1)(n −1) + (22 −2)(n −2) + · · · + ((n −1)2 −(n −1)) · 1
(13 + 23 + · · · + n3) −(12 + 22 + · · · + n2)
is equal to</p>
Step-by-Step Solution
Key Concept: General
<p>The numerator is</p> n-1 X k=1 (k2 -k)(n -k). Using standard sum formulas, n-1 X k=1 (k2 -k)(n -k) = n(n -2)(n -1)(n + 1) 12 . Also, n X k=1 k3 - n X k=1 k2 = n(n -1)(n + 1)(3n + 2) 12 . Therefore, n(n-2)(n-1)(n+1) 12 n(n-1)(n+1)(3n+2) 12 = n -2 3n + 2. Hence, lim n\to \infty n -2 3n + 2 = 1<p><strong>3</strong>: </p> So the correct choice is option (2) . Shortcut / Fast View Whenever both numerator and denominator are polynomial sums, convert them to closed forms first. The limit then becomes a simple rational limit.
Correct Answer: (2)