Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Number and Algebra | Topic 1.2
📋 Learning Intentions
- Identify whether a sequence is arithmetic by checking for a constant common difference between consecutive terms.
- Find the nth term of an arithmetic sequence using the formula \( u_n = u_1 + (n-1)d \) .
- Calculate the sum of an arithmetic series using \( S_n = \frac{n}{2}(2u_1 + (n-1)d) \) or \( S_n = \frac{n}{2}(u_1 + u_n) \) .
- Apply arithmetic sequence and series formulas to real-world contexts including financial planning, construction, and scheduling.
- Determine unknown parameters (first term, common difference, number of terms) given partial information about a sequence or series.
- Use the GDC to verify calculations and solve equations arising from arithmetic sequence problems.
Real-World Introduction
A construction company is building a triangular arrangement of seats in an outdoor amphitheatre. The first row has 12 seats, the second row has 15 seats, the third row has 18 seats, and so on. The manager needs to know how many seats are in the 20th row, and the total number of seats across all 20 rows, in order to price tickets and plan staffing. This is a problem that arithmetic sequences and series solve directly.
Arithmetic patterns appear wherever quantities increase or decrease by a fixed amount: monthly savings plans, depreciation of assets by a fixed dollar amount each year, stacking objects in layers, and scheduling tasks at regular intervals. The mathematics of arithmetic sequences provides exact answers to these planning problems without requiring a spreadsheet for every scenario.
🏗️ Why This Matters Arithmetic sequences model any situation with a constant rate of change — from a salary that increases by a fixed amount each year, to the number of bricks in each row of a wall, to the distance covered in each lap of a training programme. Mastering this topic gives students a direct tool for financial modelling, logistics, and data analysis throughout the IB course.
Core Concepts & Definitions
Consider the sequence: 3, 7, 11, 15, 19, … Each term is obtained by adding 4 to the previous term. This constant addition is what makes the sequence arithmetic. The key question is: can we find any term directly, without listing every term before it?
📖 Key Definition — Arithmetic Sequence An arithmetic sequence is a