Zadanie 1g, Paper 3, maj 2026, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ2, Paper 3 (zadania wzorowane), pytanie 1(g). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
An energy-buffer simulation has integer charge levels 0,1,…,w. While the charge is strictly between 0 and w, each independent tick increases it by one with probability p and decreases it by one with probability q=1−p, where 0<p<1 and p≠q. The simulation stops as soon as charge reaches 0 (shutdown) or w (full charge). Let Rₙ be the probability of reaching w before 0, starting at charge n. Thus R₀=0, R_w=1, and Rn = qRn−1 + pRn+1 for 1 ≤ n ≤ w − 1.
Polecenie
(i) By adding successive differences, or otherwise, prove . (ii) Given R1 = [1 − q/p]/[1 − (q/p)w], deduce Rn = [1 − (q/p)n]/[1 − (q/p)w].
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Odpowiedź z klucza
(i) Add differences from k=1 to n: . Substituting 1f gives , which is the claimed sum. Equivalently sum from k=2, telescope to Rₙ−R₁, then add R₁. (ii) Since q/p≠1, the finite geometric sum is [1−(q/p)^n]/[1−q/p]. Multiply by the given R₁ and cancel the nonzero factor 1−q/p to obtain the claimed ratio. Direct multiplication of (1−q/p) by the finite sum also telescopes and proves the result.
Schemat punktowania
(i) A1: correctly express the successive differences as powers times R₁. A1: identify their telescoping sum (Rₙ−R₁ if starting at k=2, or Rₙ if starting at k=1). M1: equate the two expressions. A1: correctly rearrange/factor to obtain the sum. An equivalent four-mark route explicitly writes at least two differences, uses R₀=0, gives the full telescoping identity and substitutes the geometric terms. (ii) M1: recognize the first term and ratio of the geometric sum. A1: its correct finite sum. M1: substitute the given R₁. A1: correct cancellation establishing the ratio. Alternatively award A1 for substituting R₁ into the unsummed series, M1 for multiplying by 1−q/p, and A1 each for correct initial and final cancellation terms.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact values or three significant figures unless requested otherwise; proofs and numerical solutions require supporting work.