Finite series — A/L Combined Mathematics

What must I remember about Finite series?

• Sequence vs Series: A sequence is an ordered progression of terms u1, u2, u3, ..., un; a Series is the sum of terms of a sequence: Sn = ∑_r=1ⁿ ur.
• Properties of Sigma notation: ∑_r=1ⁿ c = n*c (for constant c); ∑_r=1ⁿ c*ur = c * ∑_r=1ⁿ ur; ∑_r=1ⁿ (ur ± vr) = ∑ ur ± ∑ vr; index shifting: ∑_r=1ⁿ ur = ∑_r=0^n-1 u_r+1.
• Arithmetic Progression (A.P.): u_r = a + (r - 1)d with common difference d = u_r - u_r-1; sum of first n terms: Sn = (n/2)[2a + (n - 1)d] = (n/2)(first term + last term).
• Geometric Progression (G.P.): u_r = a * r^r-1 with common ratio r = u_r / u_r-1; sum of first n terms: Sn = a(1 - rⁿ) / (1 - r) for r ≠ 1.
• General term from sum: The r-th term of any series can be recovered from partial sums using identity: ur = Sr - S_r-1 for r ≥ 2, with u1 = S1.

Test yourself (1 minute)

1. Arithmetic Mean (AM) and Geometric Mean (GM)

   For numbers a and b: AM = (a + b)/2; GM = √(ab); for positive distinct numbers, AM > GM.

2. Harmonic Progression (H.P.)

   A sequence whose reciprocals form an A.P.; harmonic mean HM = 2ab / (a + b); connection: AM * HM = (GM)².

3. Sigma linearity limitation

   Note that ∑ (ur * vr) is NOT equal to (∑ ur) * (∑ vr); sigma does not distribute over products or quotients.

Practise Finite series questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 20

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Next in the syllabus

Where does Finite series fit in the syllabus?

Year 1 · Term 3