She closed her eyes and dreamed of limits that didn't diverge.
The first question appeared. It was a beast: Find the area bounded by the curve y = e^x sin(x), the x-axis, and the lines x = 0 and x = π.
At 4:47 AM, she reached Question 9. The final one. The “challenge” problem. ib math aa hl exam questionbank
Prove by mathematical induction that for all n ∈ ℤ⁺, Σ_{k=1}^n (k * k!) = (n+1)! – 1.
Outside, a bird started singing. The deep blue of the night sky was bleeding into a pale, anxious gray. Maya saved her work, closed the laptop, and lay back on her pillow. The questionbank was merciless—a cold, infinite engine of suffering. But tonight, for a few quiet hours, she had been its master. She closed her eyes and dreamed of limits
She clicked “Generate Random Paper.”
She set down her pen. The screen glowed with the green checkmark of the official answer. Seven out of seven. A perfect paper. At 4:47 AM, she reached Question 9
Maya laughed. It was almost elegant. The base case: n=1, 1 1! = 1, and (2)! – 1 = 1. True. The inductive step: Assume true for n. Then add (n+1) (n+1)! to both sides. Left becomes sum to n+1. Right becomes (n+1)! – 1 + (n+1)*(n+1)! = (n+1)!(1 + n + 1) – 1 = (n+2)! – 1. Done.