HKDSE Maths | Arithmetic and Geometric Series (Tutor Guide) | Term, sum, application in three moves

2026-11-04

Tutor lesson-prep guides · Secondary Mathematics

Students remember both formulas but freeze at questions asking when a total first exceeds a value. The gap is not the formula: they have not translated the words into a, d or r, and n before choosing an equation.

Objectives — Students can: (1) extract the first term a and the common difference d or ratio r, and tell n, the number of terms, from the last term; (2) move between T(n) and S(n), for example with a=3 and d=4 reaching T(20)=79 and S(20)=820; (3) set an inequality for a first-exceeds question instead of adding month by month; (4) decide when an infinite geometric series converges and evaluate it.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Algebra	Simplify 3 + 4(n – 1)	4n – 1
Indices	What is 3 to the power 6?	729
Inequalities	Solve 3 + 4(n – 1) > 50	n > 12.75

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Three-question check	Simplify, indices, inequality	Answer orally
8-20 min	Term and index	Model T(n) from a and d, stress n against the last term	Write three terms
20-32 min	Where sums come from	Reverse addition for arithmetic, subtraction trick for geometric	Derive once
32-42 min	Setting up applications	Model exceeds as an inequality	Set up independently
42-55 min	Infinite series	State the convergence condition and the formula	Classify three cases
55-60 min	Exit check	Three quick questions	Answer orally

03 Teaching script (read it out)

Scene 1: n counts terms, not the last value

The first step is always to translate the words into four letters. The first term a is the value of term one, the common difference d is any term minus the one before it, and n is how many terms there are, not the value of the last one. So with a=3 and d=4, the 20th term is T(20)=3+(20-1)x4=79: here 20 is the count and 79 is the value. Students who write n=79 for the twentieth term have swapped the two. Require a line reading a=, d=, n= before any working.

Have students say aloud that n counts terms, and re-tag whenever they slip.

Scene 2: Derive both sums once

For the arithmetic sum, write S=3+(3+4)+…+79 and the same series reversed. Every pair adds to 3+79=82, there are 20 pairs, so S=20×82/2=820. For the geometric sum, multiply S by r and subtract, which cancels every middle term and leaves a(r^n-1)/(r-1). Deriving once means a forgotten formula can be rebuilt on the spot, and it shows why r=1 needs separate treatment: the denominator vanishes, so the sum is simply na.

After the derivation, ask why r=1 cannot use the formula.

Scene 3: Applications: from which term to an inequality

When a question asks in which month the total first exceeds 1,500, set S(n) > 1500 rather than adding month by month. The result is a quadratic inequality in n; solve it, then round up because n must be a whole number. For infinite geometric series, check |r|<1 first: with a=8 and r=1/2 the sum is 16, while |r| at least 1 diverges and the formula must not be used. One sentence stating the condition usually carries the mark.

Give an infinite series with r=2 and watch whether the formula is applied blindly.

04 Question types and worked steps

Type	Example	Steps	Answer
Term	Find the nth term	Write a and d or r, use T(n)	T(20)=79
Finite sum	Sum of the first n terms	Count terms, use S(n)	S(20)=820
Application	First term exceeding a value	Set an inequality, round up	n is a whole number
Infinite sum	Convergent geometric series	Check \|r\|<1, then a/(1-r)	8 and 1/2 gives 16

Sanity check (teach this): Check the numbers: a=3, d=4, n=20 gives T(20)=3+19×4=79 and S(20)=20x(3+79)/2=820. For the geometric series a=2, r=3, n=6: T(6)=2×3^5=486 and S(6)=2(3^6-1)/(3-1)=728. The infinite series with a=8 and r=1/2 sums to 8/(1-1/2)=16. Cross-check any sum by subtracting S(n-1) from S(n): the difference must equal T(n), and if it does not, the term count is wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
With a=3 and d=4, find T(20) and S(20)	79 and 820	a, d, n written first
Geometric 2, 3, 9 and so on: find S(6)	728	Six terms counted
Can a=5, r=2 be summed to infinity?	No, \|r\| at least 1 diverges	Condition stated

06 Common mistakes and what to say

Mistake	Student writes	Say this
Term count taken as the last value	The twentieth term, so n=79	n is the count, so n is 20; 79 is T(20), not n.
Wrong count in a geometric sum	S6 adds only five terms	From term one to term six is six terms, not six minus one.
Ignoring r in an infinite series	Substituting r=2 into a/(1-r)	The denominator turns negative; the formula needs \|r\|<1.
Trial and error in applications	Adding month by month until it exceeds	Set the inequality and round up: faster and it cannot miss.

07 Homework (paper, section and question numbers)

Maths subject index — All chapters
Practice paper — Sequences: six items on terms and sums

08 Exit check (last 5 minutes)

Question	Answer	Decision
T(20) and S(20)	79 and 820	Wrong: rewrite a and d
Infinite series condition	\|r\| below 1	Cannot state it: reteach act three
First step in applications	Set an inequality	Trial and error: drill again

09 Tutor FAQ

Must the formulas be memorised?

Yes, but also derivable, so a forgotten formula can be rebuilt in seconds.

Can n be zero?

Terms start at n=1, so n is a positive integer and answers to inequalities round up.

Can r be negative?

Yes, the sum formula still holds and terms alternate in sign.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.