IB Maths AA: Sequences and Series — Tutor Lesson Plan (nth term, ratio, infinite sums)

2026-11-05

Every IB sequence question asks three things: is it arithmetic or geometric, how many terms are there, and which formula applies. This lesson ties all three to one list.

Objectives — the student can classify a sequence from its first two terms, sum an arithmetic series with n over 2 times first plus last, and state that an infinite sum exists only when the ratio is smaller than 1 in size.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Substitution	For u n = 5 + 3(n – 1), what is term 10?	32
Percentage	80 rising to 100 is an increase of what percentage?	25 percent
Absolute value	Is the size of r = -0.5 greater than 1?	No, it is 0.5

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Classify	Two lists on the board	Name the difference or the ratio
8-20 min	Arithmetic term and sum	Build u n = a + (n-1)d, then S n	Get 32 and 185
20-32 min	Geometric ratio	Find r by dividing, never by subtracting	Get 486 and 728
32-44 min	Infinite sums	Contrast convergent and divergent	Get 16
44-54 min	IB answer format	Formula line first, then substitute	Rewrite one answer
54-60 min	Exit check	Three questions marked	Finish alone

03 Teaching script (read it out)

Scene 1: Arithmetic: term first, then sum

For 5, 8, 11, 14 the common difference is 3, so u n = 5 + 3(n – 1) and the tenth term is 5 + 27 = 32.
Never add term by term to sum: S n = n over 2 times first plus last, so S 10 = 5 times (5 + 32) = 185. Write the formula line before substituting, because that is where the IB method mark sits.

Student explains why 27 is added rather than 30.

Scene 2: Geometric: the ratio comes from division

For 2, 6, 18, 54 the ratio is 6 divided by 2, which is 3, not 6 minus 2. So u n = 2 times 3 to the power n minus 1, and the sixth term is 2 times 243 = 486.
The sum is a times (r to the n minus 1) over (r minus 1), so S 6 = 2 times 728 over 2 = 728, which you can check by adding the six terms 2, 6, 18, 54, 162 and 486.
Most common misconception: students see numbers rising and reach for the arithmetic formula. Do one division before deciding.

Student explains that the ratio is later term divided by earlier term, not subtracted.

Scene 3: Infinite sums: state the condition first

For 8 + 4 + 2 + 1 the ratio is 0.5, whose size is smaller than 1, so the series converges and S infinity = 8 divided by (1 minus 0.5) = 16.
The same formula on r = 2 gives nonsense: 2 + 4 + 8 keeps growing and has no sum. So the first line of any answer is the convergence condition.

Student explains the convergence condition before using the formula.

04 Question types and worked steps

Type	Example	Steps	Answer
Find the nth term	5, 8, 11, find term 10	Find d, substitute a + (n-1)d	32
Sum an arithmetic series	Same list, find S 10	n over 2 times first plus last	185
Sum a geometric series	2, 6, 18, find S 6	Find r = 3, substitute the sum formula	728
Infinite sum	8 + 4 + 2, find S infinity	Check that r is smaller than 1, then a over 1 minus r	16

Sanity check (teach this): The arithmetic sum must equal the number of terms times the average of the ends, 10 times 18.5 = 185, which matches. Each geometric term divided by 3 returns the previous one. An infinite sum cannot exceed twice the first term here, and 16 does not.

05 Independent practice (3 questions)

Question	Answer	What to watch
u n = 4 + 6(n – 1), find term 8 and the sum of 8 terms	46 and 200	Whether the formula line is written first
3, 9, 27, find term 5 and the sum of 5 terms	243 and 363	Ratio found by division
12 + 6 + 3, find the infinite sum	24	Convergence condition stated first

06 Common mistakes and what to say

Mistake	Student writes	Say this
Treating a geometric list as arithmetic	d = 6 – 2 = 4, so term 6 = 22	Say: numbers getting bigger does not mean adding the same amount, divide the later term by the earlier one first.
Forgetting the half in the sum	S 10 = 10 times 37 = 370	Say: n over 2, because ten terms average the first and last, then halve.
Miscounting the terms	From term 1 to term 10 treated as 9 terms	Say: n minus 1 counts gaps, the number of terms is 10, do not mix them.
Using the infinite formula blindly	Substituting r = 2 into a over 1 minus r	Say: if the ratio is bigger than 1 the series diverges, write the condition before the formula.

07 Homework (paper, section and question numbers)

IB maths chapter index — Use the three practice sets linked there for homework checking

08 Exit check (last 5 minutes)

Question	Answer	Decision
5, 9, 13, find term 12	49	One wrong means reteach the nth term
2, 10, 50, find term 4	250	One wrong means reteach ratio by division
20 + 10 + 5, find the infinite sum	40	Two wrong means return to act three

09 Tutor FAQ

Is this examined at both SL and HL?

Yes, both, and HL adds induction and harder summation techniques on top of this base.

Are the formulae given?

The booklet gives both sum formulae, but the convergence condition must be written by the student.

How many terms must be memorised?

None, the skill is writing the general term straight from the first two terms.

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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.