IB Maths AI SLTopic 1 ā Sequences & SeriesPaper 1 & 2Foundation note~5 min read
Language of Sequences & Series
A sequence is an ordered list of numbers; a series is what you get when you add them up. Three pieces of notation do all the work: un (the nth term), the formula for un, and Sn (the sum of the first n terms).
š What you need to know
Sequence: an ordered list of numbers ā e.g. 3, 7, 11, 15, ā¦
Terms: each number in the sequence, labelled u1, u2, u3, ā¦
nth term un: a formula that lets you find ANY term by substituting its position n.
Series: the sum of terms in a sequence ā e.g. 3 + 7 + 11 + 15 + ā¦
Sn: the sum of the first n terms. So S4 = u1 + u2 + u3 + u4.
Position n is a positive integer (1, 2, 3, ā¦) ā never 0 or a fraction.
Sequences ā the un formula
The idea
Substitute the position n ā get the value un
Each position n (orange) maps to one term value un (teal). Sn is the sum of those values.
Series ā the sum Sn
Sum of the first n termsSn = u1 + u2 + u3 + ā¦ + un
Examples: for the sequence 3, 7, 11, 15, 19, ā¦
S1 = 3
S2 = 3 + 7 = 10
S3 = 3 + 7 + 11 = 21
S4 = 3 + 7 + 11 + 15 = 36
š§ Recipe ā working with sequences & series
Read the formula: identify what un equals in terms of n.
Find a specific term: substitute the position into the formula (e.g. u8 means put n = 8).
Find the rule from a list: look for what’s added / multiplied / squared to go from one term to the next.
To find Sn: list the first n terms, add them.
To find which position gives a value: set un = value and solve for n.
Worked examples
WE 1
Find a specific term
A sequence is given by un = 3n + 4. Find u8.
Substitute n = 8u_8 = 3(8) + 4 = 24 + 4u8 = 28to find any term, just replace n with its position.
WE 2
First few terms and S4
For the sequence un = nĀ², list the first four terms and find S4.
A sequence starts 7, 11, 15, 19, 23, ā¦ (a) Write a formula for un. (b) Find u10.
Spot the rulestarts at 7, adds 4 each timeBuild u_nu_1 = 7 = 4(1) + 3u_2 = 11 = 4(2) + 3u_n = 4n + 3Find u_10u_10 = 4(10) + 3 = 43un = 4n + 3, u10 = 43when terms go up by a fixed amount, un is always linear in n.
WE 4
Find which position gives a value
For the sequence un = 6n ā 2, find the value of n for which un = 100.
Set u_n = 100 and solve for n6n ā 2 = 1006n = 102n = 17n = 17 (so u17 = 100)n must come out as a positive integer ā if it doesn’t, 100 isn’t actually a term in the sequence.
WE 5
Real-world ā Maria’s savings
Maria saves $50 in week 1 and $5 more each week than the week before. (a) Write a formula for un, the amount saved in week n. (b) How much does she save in week 12?
List the first 6 termsu_1 = 2, u_2 = 4, u_3 = 8u_4 = 16, u_5 = 32, u_6 = 64Add themS_6 = 2 + 4 + 8 + 16 + 32 + 64S6 = 126this is a geometric sequence ā there’s a quicker formula coming in a later note.
š” Top tips
Always check u1 when writing a formula ā substitute n = 1 and confirm you get the first term.
List a few terms when you’re unsure of the rule ā patterns usually jump out by the 3rd or 4th term.
Use the GDC’s sum or list features to compute Sn for long sums.
“How many terms?” questions ā set un = the value, solve for n.
ā Common mistakes
Off-by-one in the formula: for “starts at 7, adds 4”, un = 4n + 3 (not 4n + 7). Check u1 = 7.
Confusing un with Sn: un is ONE term; Sn is the SUM up to that term.
Using n = 0: positions start at 1, not 0.
Forgetting to check whether 100 is actually a term: if solving gives n = 17.5, then 100 is not in the sequence.
Up next: Sigma Notation ā a compact way to write Sn using the Ī£ symbol. Saves writing out long sums.
Need help with AI SL Sequences & Series?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
