Zadanie 1a
(i) Find T0(6). (ii) Write a simplified expression for T0(n).
Number and algebra
Umiejętności w tym dziale
Umiejętności z tego działu sprawdza 349 zadań wzorowanych na arkuszach IB z lat 2022–2026. Dział pojawia się w 52 z 53 arkuszy w bazie.
Zadania z tego działu: 349 Otwórz w wyszukiwarce z filtrami
Zadanie 1b
Use the arithmetic-sequence sum to show that T1(n) = n2 + 2n.
Zadanie 1c
Find T2(4).
Zadanie 1d
Rectangle k has width 1, spans k − 1 ≤ x ≤ k, and has height (2k + 1)2. A quadratic curve y=f(x) passes through their top-right corners. (i) State f(x). (ii) Compare areas to show . (iii) Use a similar area comparison to show…
- Materiał graficzny
Zadanie 1e
Define un = (2n + 1)3 − (2n − 1)3. The pairs for n=1,2,3,4 are (26,26), (98,124), (a,b), (386,728). Find (i) a; (ii) b.
Zadanie 1f
For un = (2n + 1)3 − (2n − 1)3, (i) suggest an expression for ; (ii) prove it using the definition of u.
Zadanie 1g
The two bounds to verify are and . Again un = (2n + 1)3 − (2n − 1)3. (i) Express u_n as a quadratic in v = 2n + 1. (ii) Hence express in terms of T2(n), T1(n), T0(n). (iii) Hence show that…
Zadanie 1e
For general 0<p<1 with p≠q, replace Rₙ by (p+q)Rₙ in the recurrence and prove Rn+1 − Rn = (q/p)(Rn − Rn−1).
Zadanie 1f
(i) Use R₀=0 to express R₂−R₁ in terms of p,q,R₁. (ii) Show that R3 − R2 = (q/p)2R1. (iii) Deduce the general expression for Rₙ−Rₙ₋₁.
Zadanie 1g
(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].