DSE Maths | Arithmetic and Geometric Sequences Lesson-Prep Guide (for tutors)

DSE Maths | Arithmetic and Geometric Sequences Lesson-Prep Guide (for tutors) | terms, sums, modelling

2026-11-02

Tutor lesson-prep guides · Secondary Maths (DSE)

Students die on one thing: they cannot tell T(n) from S(n). Every question here starts with writing whether it asks for a term or for a total.

Objectives — Students can (1) write the general term from the first term and common difference or ratio and evaluate it; (2) apply the sum formula for the first n terms; (3) translate a word problem into a, d or a, r and decide whether it asks for a term or a total.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Substitution	T(n) = a + (n-1)d with a = 3 and d = 4; find T(4)	15
Indices	3 to the power 5	243
Linear equation	2n – 1 = 19, find n	n = 10

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Diagnostic	Three quick checks on indices and substitution	Answer on the spot and call out results
5-20 min	Model both general terms	Write arithmetic and geometric terms side by side and circle the difference	Copy the formulas and label a, d, r
20-38 min	Model both sums	Compute T and S for the same sequence and compare	Follow along and check
38-52 min	Word-problem modelling	Read three sentences and extract a, d or r	Write a and d or r themselves
52-60 min	Error correction and exit check	Name four errors, ask three exit questions	Answer orally and self-assess

03 Teaching script (read it out)

Scene 1: General term: arithmetic adds, geometric multiplies

Write two lines: arithmetic T(n) = a + (n-1)d, geometric T(n) = a times r to the power (n-1).
With a = 3 and d = 4, term 10 is 3 plus 9 times 4, which is 39.
Then ask whether term 1 equals 3 and make them substitute n = 1. That single check prevents half of all slips.

Stress: the bracket is (n-1), never (n+1). Dropping the bracket is the most frequent error.

Scene 2: Sums: decide first whether previous terms are included

For a = 3 and d = 4, the sum of 10 terms is 10/2 times (2 times 3 plus 9 times 4), which is 210.
Explain it as average times count so n/2 makes sense. Geometric: a = 2, r = 3, T(6) = 486 and S(6) = 728.

Remind: for positive sequences the sum exceeds any single term, so S below T is automatically wrong.

Scene 3: Word problems: write a and d or r, then decide term or total

Three steps: the first number is a; the constant add or multiply is d or r; the word first or next means a term, the word total means a sum.
Example: 3 dollars in month one, plus 4 each month. Month 10 is 39 dollars, and all 10 months total 210 dollars. Give both so students see the difference.

Stress: underline words such as total, altogether, in all, and write S(n) beside them.

04 Question types and worked steps

Type	Example	Steps	Answer
Find a term	a = 3, d = 4, find T(10)	1 write general term 2 substitute 3 evaluate	39
Find the number of terms	a = 3, d = 4, when does it reach 99	1 set T(n) = 99 2 solve 3 answer must be a positive integer	n = 25
Find a sum	a = 3, d = 4, find S(10)	1 write sum formula 2 substitute 3 bracket first	210
Geometric application	first term 2, ratio 3, find T(6) and S(6)	1 recognise repeated multiplying 2 term formula 3 sum formula	486 and 728

Sanity check (teach this): Substituting n = 1 into the general term must return a = 3; if not, the formula is miswritten.
The arithmetic sum 210 should match first plus last, times count, divided by two: (3 plus 39) times 10 over 2.
With r = 3 each term triples, so T(6) = 486 must exceed S(5); a smaller value means the index is wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
a = 5, d = 3, find T(12)	38	did they use (n-1)
a = 4, d = 6, find S(10)	310	bracket computed first
first term 3, ratio 2, find T(8)	384	is the index 7 or 8

06 Common mistakes and what to say

Mistake	Student writes	Say this
Missing bracket	writing a + n-1d	Write a + (n-1)d. Without the bracket n = 1 already fails.
Term and sum confused	answering 210 for the month-10 amount	The question asks month 10, so the answer is T(10) = 39; 210 is the total of all ten months.
Geometric treated as arithmetic	saying 2, 6, 18 adds 4 each time	2 to 6 is times 3 and 6 to 18 is times 3, so the ratio is 3.
Wrong index start	writing T(6) = 2 times 3 to the power 6	The first term is 2 times 3 to the power 0, so term 6 is 2 times 3 to the power 5, which is 486.

07 Homework (paper, section and question numbers)

Secondary Maths (DSE) lesson-prep index — compare this ten-section structure
School S5 sequences paper — Paper 1 Q1 to Q5, general terms and number of terms
School S5 sequences paper — Paper 2 Q6 to Q10, sums and word problems

08 Exit check (last 5 minutes)

Question	Answer	Decision
Write the arithmetic general term	a + (n-1)d	wrong means rewrite the formula
a = 3, d = 4, find T(10)	39	wrong means redo act two
Which formula does total need	S(n)	wrong means reread the three modelling steps

09 Tutor FAQ

Do we need both sum formulas?

One arithmetic version is enough (count times average), but the geometric formula a(r to the n minus 1) over (r minus 1) must be memorised.

How do we speed up slow calculators?

Split the work: write the formula, substitute, then press the calculator. Fewer keystroke errors than one long chain.

When is a problem geometric?

Words such as doubles, triples, multiplied by 3, depreciates mean geometric; adds each month or increases yearly means arithmetic.

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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.