IB Maths AI HLSequences & SeriesPaper 1 & 2un & Sn~6 min read
Language of Sequences & Series
A sequence is an ordered list of numbers built from a rule; a series is what you get when you add those numbers up. This note sets out the notation — un for a term, Sn for a sum — that the rest of the chapter relies on.
📘 What you need to know
A sequence is an ordered set of numbers with a rule for generating them.
Each number is a term; terms are labelled u1, u2, u3, … and the general one is un.
A formula for un gives any term: substitute the term number n into it.
A series is the sum of the terms of a sequence.
Sn is the sum of the first n terms: Sn = u1 + u2 + … + un.
To find which term has a given value, set un equal to it and solve for n.
What is a sequence?
A sequence is an ordered set of numbers produced by a rule — for instance “start at 1 and add 2” gives 1, 3, 5, 7, … Each number is a term, and terms are labelled with the letter u and a subscript: u1 is the first term, u2 the second, and un the general nth term.
When a formula for un is given, any term is found by substituting the term number n. For example, if un = 2n − 1, then u5 = 2(5) − 1 = 9.
What is a series?
A series is formed by adding the terms of a sequence. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 + …
The notation Sn means the sum of the first n terms.
Term and sum notationun = the nth term — substitute n into the rule
Sn = u1 + u2 + u3 + … + una sequence is the list of terms; a series is their sum
The five terms u1 to u5 form the sequence; their total S5 is the corresponding series.
Finding the rule of a sequence
Sometimes you are given the terms and must find the rule yourself. Look at how each term changes — a constant step from one term to the next points to a formula of the form un = an + b.
Working backwards: to find which term equals a particular value, set the rule un equal to that value and solve the equation for n. The answer must be a positive whole number.
🧠Recipe — working with sequences and series
Read the ruleun — the formula that generates the terms.
For a single term, substitute that value of n into un.
To list terms, substitute n = 1, 2, 3, … in turn.
For a series sum, add the first n terms: Sn = u1 + … + un.
To find a term’s position, set un equal to the value and solve for n.
Worked examples
WE 1
Finding a particular term
A sequence has nth term un = 5n − 2. Find the 8th term.
the 8th term means substitute n = 8u8 = 5(8) − 2= 40 − 2u8 = 38the subscript is the term number — put it straight into the rule.
WE 2
Listing the first terms
Write down the first four terms of the sequence with un = n2 − 1.
substitute n = 1, 2, 3, 4 in turnu1 = 12 − 1 = 0u2 = 22 − 1 = 3u3 = 32 − 1 = 8, u4 = 42 − 1 = 150, 3, 8, 15the rule need not be linear — here the terms grow by larger and larger steps.
WE 3
Finding a series sum
A sequence has nth term un = 3n − 2. Find S5.
list the first five termsu1…u5 = 1, 4, 7, 10, 13S5 is the sum of those termsS5 = 1 + 4 + 7 + 10 + 13S5 = 35a series is just the running total — list the terms, then add.
WE 4
Finding the rule from the terms
A sequence begins 7, 11, 15, 19, … (a) Describe the rule. (b) Write down u5. (c) The nth term is un = an + b — find a and b.
(a) look at the step between terms7 → 11 → 15 → 19: add 4 each time(b) next term after 19u5 = 19 + 4 = 23(c) step is 4, so a = 4; u1 = 7 gives 4 + b = 7(a) start at 7, add 4 · (b) 23 · (c) un = 4n + 3the constant step is the coefficient of n; fix b by checking u1.
WE 5
Finding a term’s position
A sequence has nth term un = 6n + 5. Which term of the sequence is equal to 95?
set the rule equal to 956n + 5 = 95solve for n6n = 90 ⇒ n = 1595 is the 15th termn must be a positive whole number — if it isn’t, the value is not in the sequence.
WE 6
Full question: terms and sum
A sequence has nth term un = 9 − 3n. (a) Find the first five terms. (b) Find S5.
(a) substitute n = 1 to 5u1 = 9 − 3 = 6, u2 = 3, u3 = 0u4 = 9 − 12 = −3, u5 = −6(b) add the five termsS5 = 6 + 3 + 0 + (−3) + (−6)(a) 6, 3, 0, −3, −6 · (b) S5 = 0a negative coefficient of n makes a decreasing sequence — the sum can be zero.
💡 Top tips
The subscript on u is the term number — u8 means substitute n = 8.
Keep sequence and series separate: a list of terms versus their sum.
Sn always means the sum of the first n terms, starting from u1.
To find a rule, check the step between terms — a constant step gives un = an + b.
When solving un = value for n, the answer must be a positive integer.
âš Common mistakes
Confusing the term number with the term value — u5 is the 5th term, not the number 5.
Mixing up sequence and series — a series is the sum, not the list.
Starting Sn from the wrong term — it always begins at u1.
Dropping negative terms when adding a series — include their sign.
Accepting a non-integer n — if n is not a positive whole number, the value is not a term.
Next up: Sigma Notation — a compact way to write a series using the symbol Σ. It is the same idea as Sn, just written more efficiently, with limits telling you where the sum starts and stops.
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