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IB DP Maths AA · Topic 1 Number and algebra · SL 1.2, 1.3, 1.8, 1.9 · AHL 1.10 · ~8 min read

Sequences, series and the binomial theorem

Arithmetic and geometric sequences and series, sigma notation, sums to infinity, the binomial theorem with Pascal's triangle and ⁿCᵣ, and (HL) the binomial series for negative and fractional powers.

Arithmetic sequencesGeometric sequencesSigma notationSum to infinityBinomial theoremⁿCᵣBinomial series (HL)
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • An arithmetic sequence adds the same d each time; a geometric sequence multiplies by the same r. Each has a formula for the nth term and for the sum of n terms, all in the formula booklet.
  • A geometric series has a sum to infinity u1/(1 − r) only when |r| < 1.
  • The binomial theorem expands (a + b)n using the coefficients ⁿCr (Pascal's triangle). The general term finds any single term without expanding everything.
  • HL: for negative or fractional n, (1 + x)n becomes an infinite series, valid only for |x| < 1.
  • 2029 syllabus: financial applications (compound interest, annuities) are removed from AA. Everything else here is unchanged.

By the end you can

  • Use the nth term and sum formulae for arithmetic and geometric sequences.
  • Use sigma notation and solve problems for an unknown n.
  • Find and use the sum to infinity of a convergent geometric series.
  • Expand (a + b)n and find specific terms with ⁿCr.
  • Use the binomial series for any rational n and state its validity. HL

1Arithmetic sequences and series

nuₙ1510arithmetic: +12 each timegeometric: ×1.8 each timeArithmetic terms lie on a straight line; geometric terms grow (or shrink) by a constant ratio.
Plot the terms and the difference is obvious: a straight line (arithmetic) or a curve (geometric).
un = u1 + (n − 1)d FORMULA BOOKLET
nth term
Sn = n2(2u1 + (n − 1)d) = n2(u1 + un) FORMULA BOOKLET
sum of the first n terms
Worked example

An arithmetic sequence has u1 = 7 and d = 4. Find u20 and S20.

  1. u20 = 7 + 19 × 4 = 83 A1
  2. S20 = 202(7 + 83) = 900 M1A1
Worked example

For 3 + 7 + 11 + …, find the smallest n with Sn > 500.

  1. Sn = n2(6 + 4(n − 1)) = n(2n + 1) M1
  2. 2n2 + n − 500 > 0 ⇒ n > 15.6 (GDC or quadratic formula) M1
  3. n = 16 A1
Common mistakeGiving n = 15.6. n counts terms, so it must be a positive integer: round up for "exceeds" questions.

2Geometric sequences and series

un = u1rn−1 FORMULA BOOKLET
nth term
Sn = u1(rn − 1)r − 1 = u1(1 − rn)1 − r, r ≠ 1 FORMULA BOOKLET
sum of the first n terms
Worked example

A geometric sequence starts 3, 6, 12, … Find u10 and S10.

  1. r = 2 A1
  2. u10 = 3 × 29 = 1536 A1
  3. S10 = 3(210 − 1)/(2 − 1) = 3069 A1
Worked example

The 2nd term of a geometric sequence is 12 and the 5th is 96. Find u1 and r.

  1. u1r = 12 and u1r4 = 96 M1
  2. divide: r3 = 8, so r = 2 A1
  3. u1 = 6 A1
Exam tip"Hence" after finding r usually means: substitute it straight into a sum formula. Divide one term equation by another to eliminate u1.

Sum to infinity

½¼⅛01½ + ¼ + ⅛ + … never passes 1: S∞ = u₁/(1 − r) = ½/(1 − ½) = 1An infinite geometric series converges when |r| < 1If |r| ≥ 1 the terms do not shrink, and the sum grows without limit (diverges).Sₙ = u₁(1 − rⁿ)/(1 − r): as n → ∞, rⁿ → 0 when |r| < 1.
Each bar fills half the gap that is left, so the total approaches 1 but never exceeds it.
S∞ = u11 − r, |r| < 1 FORMULA BOOKLET
sum to infinity of a convergent geometric series
Worked example

Find the sum to infinity of 24 + 12 + 6 + … and write 0.272727… as a fraction.

  1. r = ½: S∞ = 24/(1 − ½) = 48 A1
  2. 0.27 + 0.0027 + …: u1 = 0.27, r = 0.01 M1
  3. S∞ = 0.27/0.99 = 311 A1
Worked example

For which x does 1 + 2x + 4x2 + … converge, and what is its sum?

  1. r = 2x, so we need |2x| < 1, i.e. −½ < x < ½ R1A1
  2. S∞ = 11 − 2x A1
Common mistakeUsing S∞ when |r| ≥ 1. Always state "|r| < 1, so the series converges" before using the formula: it is often an R1 mark.

3Sigma notation

  • Σr=1n f(r) means: substitute r = 1, 2, …, n and add. The number of terms is (top − bottom + 1).
Worked example

Evaluate Σr=110 (3r + 2).

  1. it is arithmetic: first term 5, last term 32, 10 terms M1
  2. sum = 102(5 + 32) = 185 A1
GDC (calculator) methodPaper 2: use the sum(seq(…)) function (TI) or the Σ template (TI-Nspire, Casio) to evaluate sigma expressions directly, and a table of values to find the first n where a sum passes a target.
Syllabus noteFinancial applications (compound interest, depreciation, loans and annuities with the TVM solver) are removed from AA for the 2029 syllabus. Students sitting exams up to November 2028 still need them.

4The binomial theorem

111121133114641151010511615201561Each entry is the sum of the two above. Row n gives ⁿCᵣ, the coefficients of (a + b)ⁿ.
Row 5 (orange) gives the coefficients of (a + b)5: 1, 5, 10, 10, 5, 1.
(a + b)n = an + ⁿC1an−1b + … + ⁿCran−rbr + … + bn FORMULA BOOKLET
n ∈ ℕ
ⁿCr = n!r!(n − r)! FORMULA BOOKLET
the number of ways of choosing r items from n
  • General term: Tr+1 = ⁿCran−rbr. Powers of a go down, powers of b go up, and they always add to n.
Worked example

Find the coefficient of x3 in (2x − 3)5.

  1. the term with x3 is ⁵C2(2x)3(−3)2 M1
  2. = 10 × 8x3 × 9 A1
  3. coefficient 720 A1
Worked example

Find the constant term in (x + 2x2)6.

  1. general term ⁶Crx6−r(2x−2)r = ⁶Cr2rx6−3r M1
  2. constant: 6 − 3r = 0 ⇒ r = 2 A1
  3. ⁶C2 × 22 = 60 A1
Common mistakeForgetting to raise the coefficient: (2x)3 = 8x3, not 2x3. Put brackets round each part before you raise it.

5The binomial series HL

(1 + x)n = 1 + nx + n(n − 1)2!x2 + n(n − 1)(n − 2)3!x3 + … FORMULA BOOKLET
any rational n; infinite series, valid for |x| < 1
  • Not of the form (1 + x)? Factor first: (a + bx)n = an(1 + bax)n, valid for |bx/a| < 1.
Worked example

Expand √(4 + x) up to x2 and state where it is valid.

  1. √(4 + x) = 2(1 + x4)1/2 M1
  2. = 2(1 + ½(x4) − ⅛(x4)2 + …) A1
  3. = 2 + x4 − x264 + …, valid for |x| < 4 A1A1
Worked example

Use the expansion to estimate √4.2.

  1. x = 0.2: 2 + 0.05 − 0.000625 M1
  2. ≈ 2.04938 (calculator: 2.04939) A1
Linking questionGeometric series reappear in the Maclaurin series note, where ex and ln(1 + x) become infinite series, and in proof by induction, which can prove the sum formulae themselves.
Idea for your IA / explorationExplore the convergence of different series (geometric, harmonic, Σ1/r2) and how fast their partial sums approach the limit; or model a real loan or a bouncing ball with a geometric series.

Test yourself — cover the answers first

Quick check

  1. Find u12 for 5, 8, 11, …
  2. Find S8 for 2, 6, 18, …
  3. Does 9 − 6 + 4 − … converge? Find S∞ if so.
  4. Evaluate ⁷C3.
  5. How many terms are in Σr=420 r2?
  6. HL For which x is the expansion of (1 − 3x)−1 valid?

Exam-style

  1. The sum of the first n terms of an arithmetic sequence is Sn = 3n2 + 2n. Find u1 and d.
  2. A ball is dropped from 2.0 m and each bounce reaches 75 % of the previous height. Find the total distance travelled.
  3. HL Find the first three terms of (1 + 2x)−2 and use them to estimate 1/1.022.
Show answers
  1. 5 + 11 × 3 = 38
  2. 2(38 − 1)/2 = 6560
  3. r = −⅔, |r| < 1: S∞ = 9/(5/3) = 5.4
  4. 35
  5. 17
  6. |x| < ⅓
  7. u1 = S1 = 5
    u2 = S2 − S1 = 16 − 5 = 11
    d = 6
  8. down 2.0, then up and down 2 × (1.5 + 1.125 + …)
    = 2.0 + 2 × 1.5/(1 − 0.75)
    = 14 m
  9. 1 − 4x + 12x2 − …
    x = 0.01: 1 − 0.04 + 0.0012
    ≈ 0.9612
Practise this chapter
IB Maths AA practice papers
Next chapterExponents and logarithms →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.