Cambridge IGCSE Additional Mathematics Notes 12: Series

Cambridge IGCSE Additional Mathematics 0606 Topic 12 combines the positive-integer binomial theorem with arithmetic and geometric progressions. Candidates use general terms, finite sums and the convergence condition for a geometric sum to infinity. Greatest-term questions and coefficient-property investigations are excluded.

1. Sequence, progression and series

A sequence lists ordered terms. A progression follows a specified arithmetic or geometric rule. A series is a sum of terms.

The nth term answers “what is the value at position n?” while a partial sum answers “what is the total of the first n terms?” Distinguish Tₙ from Sₙ before selecting a formula.

Context determines whether the first observed quantity is term 1 or an initial term at n = 0. Define the indexing explicitly.

2. Arithmetic progressions

An arithmetic progression has constant difference d. With first term a:

Tₙ = a + (n - 1)d

Sₙ = n/2 [2a + (n - 1)d].

The sum can also be n/2(first term + last term). These formulas are supplied. A negative d produces a decreasing progression but remains arithmetic.

Use two known terms to form linear equations in a and d. Apply the position offset n - 1 carefully.

3. Arithmetic contexts

Arithmetic models suit quantities changing by a fixed amount per step, such as rows gaining the same number of seats or a monthly payment rising by a constant amount.

If asked when a term exceeds a threshold, solve an inequality in integer n and select the first valid position. If asked for total accumulated quantity, use Sₙ rather than Tₙ.

Check whether a model eventually predicts impossible negative values, which limits its contextual domain even if the algebraic sequence continues.

4. Geometric progressions

A geometric progression has constant ratio r. With first term a:

Tₙ = ar^(n - 1)

Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.

These formulas are supplied. A negative ratio alternates signs. A ratio between zero and one gives positive decay, while a ratio above one gives growth.

Use two known terms by division to eliminate a, but remember that even powers may allow positive and negative ratio cases.

5. Sum to infinity

A geometric progression has a finite sum to infinity only when |r| < 1. Then

S∞ = a/(1 - r).

The terms approach zero, and the remaining tail becomes arbitrarily small. Terms approaching zero are necessary but the geometric structure and |r| < 1 condition establish convergence here.

If |r| ≥ 1, explain that the terms do not shrink appropriately or partial sums do not settle. Do not apply the formula mechanically.