Preview
Geometric Sequences and Series
IB DP Mathematics: Applications and Interpretation | SL & HL | Number and Algebra — Topic 1.3
📋 Learning Intentions
- Identify whether a sequence is geometric by checking for a constant common ratio between consecutive terms.
- Calculate the \(n\)th term of a geometric sequence using the formula \(u_n = u_1 \cdot r^{n-1}\).
- Find the sum of a finite geometric series using \(S_n = \dfrac{u_1(r^n - 1)}{r - 1}\).
- Determine the sum to infinity of a convergent geometric series when \(|r|
- Apply geometric sequences and series to real-world contexts including compound interest, population growth, and depreciation.
- Use the GDC to solve problems involving unknown terms, ratios, and sums in geometric sequences.
Real-World Introduction
Imagine a single bacterium that doubles every hour. After one hour there are 2, after two hours there are 4, after three hours there are 8. The count grows not by adding a fixed amount each time, but by multiplying by the same factor — 2 — at every step. This multiplicative structure appears throughout nature, finance, and technology.
The same pattern governs compound interest in a savings account, the depreciation of a car's value, the spread of a viral social media post, and the decay of a radioactive substance. In each case, the quantity at each stage is a fixed multiple of the quantity at the previous stage.
🌍 Why This Matters Geometric sequences underpin compound interest calculations — the foundation of every mortgage, investment fund, and loan repayment schedule. The IB Financial Mathematics topic (Topic 1.4) is built directly on the geometric series formula, making this one of the most practically important topics in the course.
Core Concepts & Definitions
Consider the sequence \(3, 6, 12, 24, 48, \ldots\) Each term is obtained by multiplying the previous term by 2. Now consider \(100, 50, 25, 12.5, \ldots\) — each term is multiplied by \(\frac{1}{2}\). Both sequences share the same defining property: a constant multiplicative factor between consecutive terms.
📖 Key Definition — Geometric Sequence A geometric sequence is an ordered list of terms in which each term is obtained by multiplying the previous term by a fixed non-zero constant called the common ratio , \(r\).
- The first term is denoted \(u_1\).
- The common ratio is \(r = \dfrac{u_{n+1}}{u_n}\) for any consecutive pair