H2 Maths Notes (JC 1-2): 2) Sequences and Series

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 \( u_{1}, u_{2}, \dots \) defined by either an explicit formula \( u_n \) or a recurrence relation \( u_{n+1} = f(u_n) \).
• A series is the sum of the first \(n\) terms: \( S_n = u_{1} + u_{2} + \dots + u_n \).
• Use sigma notation \( S_n = \sum_{k=1}^{n} u_k \) to condense working and match marking scheme expectations.

Arithmetic progressions (AP)

• Common difference \( d = u_{n+1} - u_n \).
• General term \( u_n = u_{1} + (n - 1)d \).
• Sum \( S_n = \tfrac{n}{2} (u_{1} + u_n) \).

Geometric progressions (GP)

• Common ratio \( r = \dfrac{u_{n+1}}{u_n} \) (assuming \( u_n \neq 0 \)).
• General term \( u_n = u_{1} r^{n-1} \).
• Finite sum \( S_n = u_{1} \space \dfrac{1 - r^n}{1 - r} \) when \( r \neq 1 \).
• Infinite sum converges only if \( |r| < 1 \), giving \( S_\infty = \dfrac{u_{1}}{1 - r} \).

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 \( u_{n+1} = 1.005 u_n + 200 \) with \( u_0 = 0 \).

1. Generate the first few values on the GC using the recurrence mode to observe the growth.
2. Solve the steady-state by equating \( u_{n+1} = u_n = L \), giving \( L = 1.005 L + 200 \Rightarrow L = \frac{200}{0.005} = 40000 \).
3. Interpret: the balance approaches 40000 dollars as \(n\) becomes large.

Example 2 -- Sum to infinity

Given the GP \(5, 2.5, 1.25, \dots\), determine \( S_\infty \).

1. \( u_{1} = 5 \), \( r = 0.5 \).
2. Since \( |0.5| < 1 \), the sum converges.
3. \( S_\infty = \frac{5}{1 - 0.5} = 10 \).

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.

1. Let the common starting value be \(a\).
2. AP terms: \( a \), \( a + 3d \), \( a + 6d \). GP terms: \( a \), \( ar \), \( ar^2 \).
3. Setting \( a + 3d = ar \) and \( a + 6d = ar^2 \) gives two equations in \(r\).
4. Eliminate \(d\) to obtain \( r = 2 \), hence \( d = \frac{a}{3} \).
5. Conclude the AP is \( a, \tfrac{4a}{3}, \tfrac{5a}{3}, \dots \).

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 \( S_n \) 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 \( S_\infty \).
• 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 \( u_{n+1} > u_n \).
• Combine AP and GP formulas carefully-write down both before substituting values.

2.5 | Quick Revision Checklist

• [ ] Translate context into explicit or recurrence form fluently.
• [ ] Compute \( S_n \) and \( S_\infty \) 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.

Next step: expand Topic 2 into sub-topic 2.1 with deeper examples, including convergence proofs and series estimation strategies.