Recognise arithmetic and geometric progressions and understand the difference between them

What is a Series?

A series is simply the sum of the terms of a sequence. Imagine you have a list of numbers and you keep adding them one after another. The final total is the series.

Arithmetic Progression (AP) 🔢

In an AP the difference between consecutive terms is always the same.

Formulae:

• nth term: $a_n = a_1 + (n-1)d$
• Sum of first n terms: $S_n = \dfrac{n}{2}\,(a_1 + a_n)$

Analogy: Think of an AP like stepping on a staircase where each step is the same height ($d$). Every time you step up, you add the same amount.

Geometric Progression (GP) 📈

In a GP each term is found by multiplying the previous term by a constant ratio.

Formulae:

• nth term: $a_n = a_1 r^{\,n-1}$
• Sum of first n terms (r≠1): $S_n = a_1\,\dfrac{r^n-1}{r-1}$

Analogy: Think of a GP like a snowball rolling down a hill: each time it rolls, it gets bigger by the same factor ($r$). The growth is multiplicative, not additive.

Key Differences ⚖️

1. AP has a constant difference $d$ between terms.
2. GP has a constant ratio $r$ between terms.
3. AP terms grow linearly; GP terms grow exponentially.
4. AP sum formula involves arithmetic mean; GP sum involves geometric series formula.

Exam Tips for IGCSE 0606 📊

• Always identify whether the series is AP or GP before applying formulas.
• Check the first term $a_1$ and the common difference/ratio $d$ or $r$ from the given data.
• For sums, remember the correct formula: $S_n = \dfrac{n}{2}(a_1 + a_n)$ for AP and $S_n = a_1\dfrac{r^n-1}{r-1}$ for GP.
• When $r=1$ in a GP, the series is actually an AP with difference $0$.
• Practice converting between terms and sums; write a few terms out to spot patterns.