4️⃣ Introduction to Financial-Mathematics Tools — Slides

Mathematical Fundamentals · MS01-001-G

Learning Objectives

Tip

A sequence \((u_n)_{n \geq 0}\) is an ordered list of real numbers. Each term \(u_n\) is identified by its index \(n\).

Sequences model quantities that evolve step by step over time — loan balances, population counts, investment values.

Common difference \(d = u_{n+1} - u_n\) (constant).

Key Formulas

General term: \[u_n = u_0 + nd\]

Sum of first \(n+1\) terms: \[S = \frac{(n+1)(u_0 + u_n)}{2}\]

Example: a salary that increases by €500 each year is arithmetic with \(d = 500\).

Common ratio \(q = \dfrac{u_{n+1}}{u_n}\) (constant, \(q \neq 0\)).

Key Formulas

General term: \[u_n = u_0 \times q^n\]

Sum (for \(q \neq 1\)): \[S = u_0 \times \frac{1 - q^{n+1}}{1 - q}\]

Example: an investment earning 3 % per year is geometric with \(q = 1.03\).

Two savings plans, each starting at €1,000.

Plan A — arithmetic (\(d = 200\), add €200/year): \[u_5 = 1000 + 5 \times 200 = €2{,}000\]

Plan B — geometric (\(q = 1.10\), grow by 10 %/year): \[u_5 = 1000 \times 1.10^5 \approx €1{,}611\]

Conclusion: arithmetic grows faster in the short run; geometric catches up with a high ratio over time.

Arithmetic sequences add a constant difference — growth is linear.
Geometric sequences multiply by a constant ratio — growth is exponential.
Key diagnostic: if increments are constant → arithmetic; if ratios are constant → geometric.