Sequences
Numerical sequences, their general term and recurrence definitions, and the two great families: arithmetic and geometric progressions.
- \(\mathbb{N} = \{1, 2, 3, \dots\}\) is the set of natural numbers (indices of terms).
- A function's rule assigns exactly one output to each input.
4.1 Sequences — generalities and the general term
A sequence is a function \(u:\mathbb{N}\to\mathbb{R}\). We write \(u_n\) for the term of order \(n\). The formula giving \(u_n\) in terms of \(n\) is the general term.
If \(u_n = 2n + 1\), then \(u_1 = 3,\ u_2 = 5,\ u_3 = 7,\dots\) (the odd numbers from 3 upward).
A sequence is increasing if \(u_{n+1} > u_n\) for all \(n\), decreasing if \(u_{n+1} < u_n\), and bounded if all its terms lie within a fixed interval.
4.2 Graphical representation
Because the domain is \(\mathbb{N}\), the graph of a sequence is a set of isolated points \((n, u_n)\) — never a continuous curve. This mirrors the dotted step pattern shown on the book cover.
4.3 Sequences defined by recurrence
A recurrence gives each term from previous ones, together with a starting value. Example:
\[ \begin{cases} u_1 = 2 \\ u_{n+1} = u_n + 3 \end{cases} \Rightarrow 2, 5, 8, 11, \dots \]The famous Fibonacci sequence is a recurrence needing two seeds: \(f_1 = f_2 = 1,\ f_{n+2} = f_{n+1} + f_n\).
4.4 Arithmetic progressions
An arithmetic progression (AP) has a constant difference \(r\) between consecutive terms: \(u_{n+1} - u_n = r\).
General term: \(\displaystyle u_n = u_1 + (n-1)\,r\).
Sum of the first \(n\) terms: \(\displaystyle S_n = \frac{(u_1 + u_n)}{2}\cdot n\).
For the AP \(3, 7, 11, \dots\) (\(u_1 = 3,\ r = 4\)): \(u_{10} = 3 + 9\cdot 4 = 39\), and \(S_{10} = \dfrac{3 + 39}{2}\cdot 10 = 210.\)
4.5 Geometric progressions
A geometric progression (GP) has a constant ratio \(r\) between consecutive terms: \(\dfrac{u_{n+1}}{u_n} = r\), for all \(n\), with \(u_n \neq 0\).
General term: \(\displaystyle u_n = u_1 \cdot r^{\,n-1}\).
Sum of the first \(n\) terms (\(r \neq 1\)): \(\displaystyle S_n = u_1 \cdot \frac{1 - r^{\,n}}{1 - r}\).
For the GP \(2, 6, 18, \dots\) (\(u_1 = 2,\ r = 3\)): \(u_5 = 2\cdot 3^4 = 162\), and \(S_5 = 2\cdot \dfrac{1 - 3^5}{1 - 3} = 2\cdot \dfrac{-242}{-2} = 242.\)
Practice problems
- Write the first four terms of \(u_n = \dfrac{n}{n+1}\).
Show solution
\(\tfrac12, \tfrac23, \tfrac34, \tfrac45\). - An AP has \(u_1 = 5\) and \(r = -2\). Find \(u_{20}\) and \(S_{20}\).
Show solution
\(u_{20} = 5 + 19(-2) = -33\). \(S_{20} = \tfrac{5 + (-33)}{2}\cdot 20 = -280\). - A GP has \(u_1 = 1\) and \(r = \tfrac12\). Find \(u_6\) and \(S_6\).
Show solution
\(u_6 = (\tfrac12)^5 = \tfrac{1}{32}\). \(S_6 = \dfrac{1 - (1/2)^6}{1 - 1/2} = 2\left(1 - \tfrac{1}{64}\right) = \tfrac{63}{32}\). - Is the sequence \(u_n = 3^n\) arithmetic, geometric, or neither?
Show solution
\(\tfrac{u_{n+1}}{u_n} = 3\) constant — geometric with ratio 3.
Summary
- A sequence is a function on \(\mathbb{N}\); its graph is isolated points.
- AP: constant difference \(r\); \(u_n = u_1 + (n-1)r\); \(S_n = \tfrac{(u_1+u_n)}{2}n\).
- GP: constant ratio \(r\); \(u_n = u_1 r^{n-1}\); \(S_n = u_1\tfrac{1-r^n}{1-r}\).
- Find the 8th term of the AP \(1, 4, 7, \dots\)
- Find the sum \(2 + 4 + 8 + \dots + 256\) (a GP).