Sequences and Series

A sequence is an ordered list of numbers following a rule. A series is the sum of the terms of a sequence. Paper 01 allocates three items to this section; Paper 02 frequently integrates sequences into multi-part algebra questions.

Notation¶

The terms of a sequence are written $u_1, u_2, u_3, \ldots, u_n$, where $u_n$ is the general term (also called the $n$th term). The sum of the first $n$ terms is written $S_n$.

Sigma notation compresses series:

$S_n = \sum_{r=1}^{n} u_r = u_1 + u_2 + \cdots + u_n$

The letter $r$ is the index of summation. Key rules: $\sum (u_r + v_r) = \sum u_r + \sum v_r$ and $\sum c\,u_r = c\sum u_r$.

Arithmetic Sequences¶

An arithmetic sequence has a constant difference $d$ between consecutive terms.

$u_1,\quad u_1 + d,\quad u_1 + 2d,\quad \ldots$

The $n$th term is:

$u_n = a + (n-1)d$

where $a = u_1$ is the first term and $d$ is the common difference (which may be negative).

Identifying arithmetic sequences: check that $u_2 - u_1 = u_3 - u_2 = d$ (constant).

Sum of an Arithmetic Series¶

$S_n = \frac{n}{2}[2a + (n-1)d]$

An equivalent form when the last term $l = u_n$ is known:

$S_n = \frac{n}{2}(a + l)$

Divergence of Arithmetic Series¶

Every arithmetic series diverges unless $d = 0$. As $n$ grows without bound, $S_n$ grows without bound (positively or negatively). There is no finite sum to infinity for an arithmetic series.

Geometric Sequences¶

A geometric sequence has a constant ratio $r$ between consecutive terms.

$u_1,\quad u_1 r,\quad u_1 r^2,\quad \ldots$

The $n$th term is:

$u_n = ar^{n-1}$

Identifying geometric sequences: check that $u_2/u_1 = u_3/u_2 = r$ (constant).

Sum of a Geometric Series¶

For $r \neq 1$:

$S_n = \frac{a(r^n - 1)}{r - 1} \qquad \text{(use when } r > 1\text{)}$

$S_n = \frac{a(1 - r^n)}{1 - r} \qquad \text{(use when } |r| < 1\text{, avoids negatives)}$

Both forms are equivalent. Choose whichever keeps the numerator positive to reduce sign errors.

Convergence and Sum to Infinity¶

A geometric series converges (has a finite sum to infinity) if and only if $|r| < 1$.

When $|r| \geq 1$, the terms do not approach zero and the series diverges.

For a convergent geometric series:

$S_\infty = \frac{a}{1 - r} \qquad (|r| < 1)$

Finding the Common Ratio from a Convergence Condition¶

Comparing Arithmetic and Geometric¶

Real-World Applications¶

Repeated percentage change produces a geometric sequence. A quantity starting at $A$ and changing by a fixed factor $k$ each period has value $Ak^{n-1}$ after $(n-1)$ periods.