Zadanie 1d, Paper 3, listopad 2022, Matematyka AA IB HL

(i)$F_1\prime(x)=\sum(2i-1)^2x^{2i-2}$, hence multiplication by x gives F₂. (ii) For q=1, differentiating F₀ and multiplying by x gives the claimed formula. Assume it holds for q=k. Then $F_{k+1}=xF_k\prime=x\sum(2i-1)^{k+1}x^{2i-2}=\sum(2i-1)^{k+1}x^{2i-1}$. The base and successor step prove the statement for every positive integer q. (iii)$F_q(1)=\sum_{i=1}^n(2i-1)^q$.

(i) A1:correct general derivative term. A1:multiply by x to obtain the target; at most one mark without a general term. (ii) R1:verify q=1. M1:state an induction hypothesis. M1:apply the defining recurrence at k+1. A1:correct differentiated sum. A1:correct successor expression. R1:valid induction conclusion, requiring at least three of the preceding five marks; subsequent work is not automatically lost if the hypothesis is poorly stated. (iii) A1:correct sigma expression. In (i), also accept the product-rule route F2=xF0′+x²F0″, with one accuracy mark for the differentiated expression and one for combining its general terms. Except for the stated no-general-term cap, the second A1 depends on the first. In (ii), merely writing “let q=k” is not an induction hypothesis.