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.
She closed her eyes and dreamed of limits that didn't diverge.
She checked the solution bank. Correct. A tiny, fragile smile.
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. ib math aa hl exam questionbank
The second question was a nightmare dressed in vectors. Line L1 passes through (1,2,3) with direction (2, -1, 2). L2 is given by (x-3)/2 = (y+1)/1 = (z-4)/-2. Find the shortest distance between L1 and L2. Maya groaned. This was the kind of problem that separated the 6s from the 7s. She sketched the cross product of the direction vectors, found a vector connecting the two lines, and then did the scalar projection. Her arithmetic was shaky—she forgot a negative sign halfway through, had to erase four lines, and nearly threw her pencil across the room.
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. The final one. The “challenge” problem. Maya laughed
By the fourth question—a probability distribution with a hidden binomial and a condition that required Bayes’ theorem—she wasn't just solving. She was reading . She saw the trap before she stepped in it. The questionbank had trained her. She knew that when they said “at least two,” they meant “1 minus the probability of zero and one.” She knew that when they gave a complex number in polar form and asked for the least positive integer n such that z^n was real, they were really asking about the argument modulo π.
She clicked “Generate Random Paper.”
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. = 1, and (2)
But she finished. And the solution bank said “Correct.” Her heart beat a little faster.
Prove by mathematical induction that for all n ∈ ℤ⁺, Σ_{k=1}^n (k * k!) = (n+1)! – 1.