Prove by mathematical induction that for all n ∈ ℤ⁺, Σ_{k=1}^n (k * k!) = (n+1)! – 1.
“Okay,” she whispered, pulling out a fresh sheet of paper. “Integration by parts. Twice. Then a trick.” Her pen flew, sketching the cyclic dance of derivatives. sin(x) becomes cos(x) becomes -sin(x) . e^x stays e^x . She wrote the lines, the u and dv, the careful subtraction. Ten minutes later, she had an answer: (e^π + 1)/2 . ib math aa hl exam questionbank
But she finished. And the solution bank said “Correct.” Her heart beat a little faster. Prove by mathematical induction that for all n
She checked the solution bank. Correct. A tiny, fragile smile. “Integration by parts
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.
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.