IB Maths AI SLTopic 1 ā Sequences & SeriesPaper 1 & 2In formula booklet~7 min read
Arithmetic Sequences & Series
An arithmetic sequence goes up (or down) by the same amount each step. That fixed step is the common difference, d. Two formulae do all the work ā one for any term, one for the sum.
š What you need to know
Common difference d: subtract any term from the next one (d = u2 ā u1). Same value between every pair.
nth term formula: un = u1 + (n ā 1)d.
Sum of first n terms ā two versions, both in the formula booklet: Sn = (n/2)(2u1 + (n ā 1)d) ā use when you know u1 and d. Sn = (n/2)(u1 + un) ā use when you know u1 and un.
d > 0: sequence increases. d < 0: sequence decreases.
Given two terms, solve simultaneously for u1 and d.
Finding n from Sn gives a quadratic in n ā use your GDC’s solver.
Finding the nth term
nth term ā formula bookletun = u1 + (n ā 1)d
The common difference d is the same between every consecutive pair. To jump straight to u15 without listing all 14 terms, use the formula.
Finding the sum Sn
Sum of n terms ā both in formula bookletSn = n2(2u1 + (n ā 1)d) or Sn = n2(u1 + un)
Which version to use? If the question gives you u1 and d, use the first. If it gives you the first AND last term (u1 and un), the second is faster.
š§ Recipe ā solve any arithmetic problem
Identify u1 and d: from the sequence, or from two given terms (solve simultaneously).
To find a specific term: plug n into un = u1 + (n ā 1)d.
To find which term equals a value: set un = value and solve for n.
To find a sum: plug into the Sn formula.
To find n from Sn: get a quadratic in n, solve on GDC, take the positive integer root.
Worked examples
WE 1
Find a specific term
An arithmetic sequence has first term 7 and common difference 4. Find u15.
An arithmetic sequence has u1 = 2 and d = 3. Find the value of n for which Sn = 345.
Apply S_n formulan/2 Ć (2(2) + (n ā 1)(3)) = 345n/2 Ć (3n + 1) = 345n(3n + 1) = 690Quadratic in n3nĀ² + n ā 690 = 0Solve on GDC (Polynomial Root Finder)n = 15 or n = ā15.33ā¦ (reject negative)n = 15always reject negative or non-integer roots ā n must be a positive whole number.
WE 6
Real-world ā stadium seating
A theatre has 12 seats in the front row, and each subsequent row has 2 more seats than the row in front. How many seats are there in total across 25 rows?
Identify the arithmetic sequenceu_1 = 12, d = 2, n = 25Apply S_nS_25 = 25/2 Ć (2(12) + 24(2)) = 25/2 Ć (24 + 48) = 25/2 Ć 72S25 = 900 seatswhenever something grows by a FIXED amount each step, it’s arithmetic ā reach for these formulae.
š” Top tips
Spot arithmetic when terms go up/down by a constant ā simple interest, seats per row, salary increases by a fixed amount.
For two-term problems, subtract the equations to eliminate u1 and find d quickly.
Use GDC Polynomial Root Finder for quadratic-in-n problems ā input coefficients of the quadratic.
Always sanity-check n: must be a positive integer. Reject anything else.
ā Common mistakes
Using n instead of (n ā 1): u5 = u1 + 4d, not u1 + 5d. The 5th term has 4 jumps from the 1st.
Forgetting d can be negative: for “decreasing by 5”, d = ā5, not 5.
Mixing up the two sum formulae: the second one (u1 + un) only works if you actually KNOW un.
Keeping a non-integer or negative n: if your GDC gives n = 12.5 or n = ā8, that root is invalid.
Up next: Geometric Sequences & Series ā same idea but you MULTIPLY by a fixed amount (common ratio r) instead of adding. Same two formulae structure: one for terms, one for sums.
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