IB Maths AA HLTopic 1 ā Number & AlgebraPaper 1 & 2~7 min read
Language of Sequences & Series
Before you start crunching arithmetic and geometric series, you need to be fluent with two pieces of vocabulary that show up in every question for the rest of this section: the nth term notation un, and the partial sum notation Sn. Get these two clear and the rest of the section is just plugging into formulas.
š What you need to know
A sequence is an ordered list of numbers built by a rule. The numbers are called terms.
A series is what you get when you add up the terms of a sequence.
The nth term of a sequence is written un ā read “u sub n”. The first term is u1, the second is u2, and so on.
If you have a formula for un, you find any term by substituting that number in for n.
The sum of the first n terms is written Sn: Sn = u1 + u2 + … + un.
The variable n is always a positive integer (1, 2, 3, …) ā there’s no such thing as the 2.7th term.
What is a sequence?
A sequence is just an ordered list of numbers ā but unlike a random list, it’s built by a rule that tells you how to find every term. The rule might be “start at 4 and add 3 each time” or “double each number” or “square the position” ā anything that consistently generates the next number from what you know.
Examples:
4, 7, 10, 13, 16, … (start at 4, add 3) | 1, 4, 9, 16, 25, … (the squares) | 100, 50, 25, 12.5, … (halve each time)
In IB notation, the individual numbers in a sequence are called terms. The order matters ā 1, 3, 5 is a different sequence from 5, 3, 1, even though they contain the same numbers.
The notation ā un
To talk about specific terms without writing them all out, IB uses the letter u with a subscript indicating the position.
Naming the termsu1 = first term, u2 = second term, u3 = third term, … un = nth term (any general term)
So for the sequence 4, 7, 10, 13, 16:
u1 = 4 | u2 = 7 | u3 = 10 | u4 = 13 | u5 = 16
Some textbooks use an or tn for the same thing. IB sticks with un ā that’s what you’ll see in the formula booklet and exam papers.
Finding terms from a formula
Most exam questions give you a formula for un in terms of n. To find any specific term, just substitute that n-value into the formula.
š§ Recipe ā finding a specific term
Identify the formula for un in the question.
Substitute the term number you want into the formula in place of n.
Evaluate.
Example: If un = 3n + 1, then u1 = 3(1) + 1 = 4, u2 = 3(2) + 1 = 7, u10 = 3(10) + 1 = 31.
š¤ Two ways to define a sequence
An explicit formula (like un = 3n + 1) gives you any term directly from n. A recursive rule (like un+1 = un + 3 with u1 = 4) tells you how to get the next term from the previous one. Both define the same sequence, just from different angles. IB uses both styles.
What is a series?
A series is a sequence with plus signs between the terms. You take an ordered list of numbers and add them up.
š
Sequence (a list)
4, 7, 10, 13, 16
commas separate the terms
ā
Series (a sum)
4 + 7 + 10 + 13 + 16
a single number when totalled (= 50)
Sequence vs Series ā same numbers, different relationship
The Sn notation
Just as un names a single term, Sn names a partial sum.
Sum of the first n termsSn = u1 + u2 + u3 + … + un
So S3 means “the first three terms added together”, S10 means “the first ten terms added together”, and so on.
In Paper 1 and 2 questions you’ll often need to compute un and Sn in the same problem ā find a particular term, then sum up to it. The arithmetic and geometric notes that follow give you formulas to do this without adding everything by hand.
A heads-up on sigma notation
For longer sums, IB uses the Greek letter Ī£ (capital sigma) as shorthand. So S5 can also be written as a sum from k = 1 to 5 of uk. Sigma notation gets its own dedicated note next ā for now, just know it’s compact notation for “add these up”.
Worked examples
WE 1
Find specific terms from a formula
A sequence is defined by un = 3n + 1. Find u1, u5, and u10.
Substitute each n into the formulau1 = 3(1) + 1 = 4u5 = 3(5) + 1 = 16u10 = 3(10) + 1 = 31u1 = 4, u5 = 16, u10 = 31
WE 2
Describe the rule and find a later term
Consider the sequence 7, 10, 13, 16, 19, … (a) Describe the rule. (b) Find u8.
(a) Spot the ruleeach term is 3 more than the previous onerule: start at 7, add 3 each time(b) Continue the pattern to u8u5 = 19, u6 = 22, u7 = 25, u8 = 28u8 = 28an explicit formula here would be un = 3n + 4 ā confirms u8 = 24 + 4 = 28 ā
The first five terms of a sequence are 5, 8, 11, 14, 17. Find a formula for un.
Step 1: Spot the patterneach term is 3 more than the previous ā increases by 3 each stepStep 2: Build the formulatry un = 3n + cwhen n = 1: u1 = 3 + c = 5, so c = 2Step 3: Verifyu2 = 6 + 2 = 8 ā | u5 = 15 + 2 = 17 āun = 3n + 2
WE 5
Decreasing sequence ā terms and partial sum
A sequence is given by un = 20 ā 4n. List the first five terms and find S5.
Step 1: Compute each termu1 = 20 ā 4 = 16u2 = 20 ā 8 = 12u3 = 20 ā 12 = 8u4 = 20 ā 16 = 4u5 = 20 ā 20 = 0Step 2: Sum themS5 = 16 + 12 + 8 + 4 + 0 = 40Terms: 16, 12, 8, 4, 0 | S5 = 40terms are getting smaller because the formula has a negative coefficient on n
š” Top tips
Always start the count from n = 1 unless the question explicitly states otherwise.
Don’t confuse un with Sn. One is a single term, the other is a running total. Read the question carefully to see which is being asked for.
If a sequence has a “reverse” question (find a formula given the terms), look at how the values change from term to term ā that’s almost always how you find the coefficient on n.
For the formula un = an + b, the a is the step size and b is found by plugging in n = 1.
Substitute neatly ā keep the brackets in to avoid sign errors. un = 5 ā 2n at n = 4 is 5 ā 2(4), not 5 ā 24.
Watch for sequences where the formula uses n2, 2n, or 1n ā they’re not arithmetic, but the substitution method is the same.
For partial sums, you can always fall back on adding term by term ā but the arithmetic and geometric notes give you faster formulas.
ā Common mistakes
Confusing the term number with the term itself.u5 is the 5th term ā the value at position 5 ā not necessarily the number 5.
Counting from zero. In IB, n starts at 1, not 0. The first term is u1.
Mixing up Sn and un.S5 is “the sum of the first five terms”, not “the fifth term”.
Forgetting brackets when substituting. 20 ā 4(5) = 0, but 20 ā 45 (without brackets) is ā25. Use brackets.
Reading “5th term” as u4. The 5th term is u5 ā match the position to the subscript exactly.
Trying to use a non-integer n. There’s no u2.5. Sequences are defined only at positive integer positions.
Assuming all sequences are arithmetic. A sequence is just an ordered list with a rule ā that rule could be anything (squares, powers, alternating signs, etc.).
This vocabulary ā un, Sn, “term”, “rule” ā is the framework for the entire section. Every later note (sigma, arithmetic, geometric, applications, compound interest) uses these names without re-introducing them. Spend a few minutes making sure they feel natural before moving on.
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