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.
Geometric patterns and totals
Work with constant-ratio sequences, their finite sums and sigma notation, including growth and decay models.
Tę umiejętność sprawdza 38 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 38 Otwórz w wyszukiwarce z filtrami
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].
Zadanie 1h
The upward-tick probability is now 0.498. A run starts with m charge units and stops successfully when it has accumulated 12 additional units, so capacity is w=m+12. Find the smallest positive integer m giving at least an 85% chance of success.
Zadanie 3a
Find the common ratio r.
Zadanie 3b
Find the first term in the form a × 10k, with 1 ≤ a < 10 and integer k.
Zadanie 7a
Show that u₁=1/6.
Zadanie 7b
Find u2k+1.
Zadanie 1b
(i) State f∘4(x). (ii) Conjecture a corresponding expression for f∘n(x) for positive integers n. (iii) Use your expression, or another method, to find f∘n(x) when m = 1.
Zadanie 1c
For m ≠ 1, prove by mathematical induction that for every positive integer n.
Zadanie 5a
(i) Show that . (ii) Hence find the mean of the eight areas in the form , where p is a positive rational number and a is a positive integer.
Zadanie 5b
Find the median area in the form cm2, with q a positive rational number.
Zadanie 7a
Find the second term of the geometric sequence.
Zadanie 10b
Now set k = −6. (i) Prove that the three terms form the start of a geometric sequence. (ii) Explain why the corresponding infinite geometric series has no finite sum.
Zadanie 10c
The already-investigated geometric case is k = −6. A second value of k also makes the sequence geometric. (i) Show that k2 + 7k + 6 = 0. (ii) Find the first three terms for the second value. (iii) State the sum S2m of its first 2m terms.
Zadanie 12b
(i) Show that . (ii) Write F3(x) in ascending powers of x.
Zadanie 5b
Use the displayed cubic approximation to estimate the signal strength before storage.
Zadanie 10e
The stage crosses the reference mark at T,2T,3T,… . Let uₙ be its greatest distance from the mark during (n−1)T<t<nT. These distances form a geometric sequence. Find (i) its common ratio; (ii) the total distance travelled if the model continues indefinitely.
Zadanie 1a
Find its value at the end of the first year.
Zadanie 1b
Find its value 9 years after purchase, to the nearest dollar.
Zadanie 1c
Find the least number n of complete years after which its value is below 15% of its original price.
Zadanie 5a
Find Sₙ.
Zadanie 5b
Hence prove S1 + S2 + ⋯ + Sn = [4(4n − 1) − 3n]/9.
Zadanie 10b
Show that s2 = at.
Zadanie 10c
Show that p>2.
Zadanie 10d
Suppose a=16 and s>0. List the first four terms of (i) the arithmetic sequence; (ii) the geometric sequence.
Zadanie 1a
Verify that the start-of-year-2 estimate is 5000 pages.
Zadanie 1b
Find the start-of-year estimates for (i) year 3; (ii) year 4.
Zadanie 1c
The recurrence gives . (i) Show that Tn = 1600(9/8)n−1 + 3200. (ii) Estimate the work at the start of year 6, to the nearest whole page.
Zadanie 1d
For quota 750, Dn = −1200(9/8)n−1 + 6000. (i) Show that Dn+1 − Dn = −150(9/8)n−1. (ii) Explain why the algebraic sequence is strictly decreasing.
Zadanie 1e
Under quota 750, Dn = −1200(9/8)n−1 + 6000. At the end of which processing year does the queue first clear?
Zadanie 1f
With a general quota d, the algebraic estimate is Cn = (4800 − 8d)(9/8)n−1 + 8d. Find the quota that keeps the start-of-year work constant.
Zadanie 12b
The ordered roots form a geometric sequence. Find its common ratio in Cartesian form.
Zadanie 4
For , write (x + h)7 = x7 + ax6 + bx5 + cx4 + dx3 + ⋯ + h7. The coefficients a, b, d, in that order, form a geometric sequence. Find h.
Zadanie 10d
Find both possible common ratios of v_n.
Zadanie 10e
Given v_99<0, find v_5.
Zadanie 10a
Suppose it is geometric. (i) Show that . (ii) Show that both choices give a convergent series. (iii) For p>0 and infinite sum , find x.
Zadanie 1e
For F(x) = x + x3 + ⋯ + x2n−1, use a geometric sum to show that F(x) = x(1 − x2n)/(1 − x2) whenever x≠±1.
Zadanie 3
A geometric sequence begins with 80 and has fourth term 121.67. Let Sₙ be the sum of its first n terms. Find the least positive integer n for which Sₙ>12000.