Q: What does H2 Maths Notes (JC 1-2): 2) Sequences and Series cover? A: Arithmetic and geometric progressions, convergence tests, sigma notation, and calculator workflows for modelling recurrence relations.
Before you start Refresh IP-level arithmetic and geometric sequences. Revisit the MOE definitions so your notation matches the mark scheme and set your graphing calculator (GC) to Exam Mode before drilling.
2.1 | Language and Notation
Core ideas
A sequence is an ordered list u1,u2,… defined by either an explicit formula un
or a recurrence relation
un+1=f(un)
.
A series is the sum of the first n terms: Sn=u1+u2+⋯+un.
Use sigma notation Sn=∑k=1nuk to condense working and match marking scheme expectations.
Arithmetic progressions (AP)
Common difference d=un+1−un.
General term un=u1+(n−1)d.
Sum Sn=2n(u1+un)
Geometric progressions (GP)
Common ratio r=unun+1 (assuming un=0).
General term un=u1rn−1.
Finite sum Sn=u11−r1−rn
Infinite sum converges only if ∣r∣<1, giving S∞=1−ru1
2.2 | Worked Examples
Example 1 -- Savings plan (recurrence)
A student deposits 200 dollars monthly into an account earning 0.5 percent interest per month. The balance after n months obeys un+1=1.005un+200 with u0=0.
Generate the first few values on the GC using the recurrence mode to observe the growth.
Solve the steady-state by equating un+1=un=L, giving L=1.005L+200⇒L=0.005200=40000.
Interpret: the balance approaches 40000 dollars as n becomes large.
Example 2 -- Sum to infinity
Given the GP 5,2.5,1.25,…, determine S∞.
u1=5, r=0.5.
Since ∣0.5∣<1, the sum converges.
S∞=1−0.55=10.
Example 3 -- Mixed AP/GP question
The first, fourth, and seventh terms of an AP are equal to the first three terms of a GP. With AP common difference d and GP common ratio r, show that r=2 and find the AP.
Let the common starting value be a.
AP terms: a, a+3d, a+6d. GP terms: a, ar, ar2.
Setting a+3d=ar and a+6d=ar2 gives two equations in r
Eliminate d to obtain r=2, hence d=3a.
Conclude the AP is a,34a,35a,….
2.3 | Calculator Workflows
Use GC recurrence mode (SEQ on TI, RECUR on Casio) to generate terms quickly and observe convergence.
Employ the summation feature to evaluate Sn without manual addition, but record the command in your working (e.g. sumSeq(200(1.005)^(X-1), X, 1, 24)).
For sigma notation, store n as a variable and test values so you can spot algebraic errors before finalising.
2.4 | Exam Watch Points
Always state convergence conditions ∣r∣<1 before using S∞.
In recurrence modelling, quote the balance to the nearest cent and interpret the result (e.g. long-run savings, population cap).
Justify inequalities when proving monotonic sequences un+1>un.
Combine AP and GP formulas carefully-write down both before substituting values.
Practice Quiz
Consolidate AP/GP identities, convergence tests, and sigma manipulations before moving to sub-topic drills.
2.5 | Quick Revision Checklist
Translate context into explicit or recurrence form fluently.
Compute Sn and S∞ with and without the GC.
Explain what convergence means in words (e.g. limiting value for savings).
Manipulate sigma notation expressions by splitting or re-indexing the sum.
