Sequences and Series (Arithmetic & Geometric)

Arithmetic Sequences

An arithmetic sequence has a common difference $d$ between consecutive terms. If the first term is $u_1$, the sequence looks like:

$u_1, \; u_1 + d, \; u_1 + 2d, \; u_1 + 3d, \; \ldots$

The general (nth) term is given by:

$u_n = u_1 + (n - 1)d$

This formula is in your IB formula booklet. To use it, you need any two of $u_n$, $u_1$, $n$, and $d$ to find the others.

The sum of the first $n$ terms of an arithmetic series is:

$S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$

The second form is often more convenient when you already know the last term $u_n$. Conceptually, $S_n$ equals the number of terms multiplied by the average of the first and last terms — this makes intuitive sense and is worth understanding rather than just memorising.

Geometric Sequences

A geometric sequence has a common ratio $r$ between consecutive terms. Starting from $u_1$:

$u_1, \; u_1 r, \; u_1 r^2, \; u_1 r^3, \; \ldots$

The general (nth) term is:

$u_n = u_1 \cdot r^{n-1}$

The sum of the first $n$ terms of a geometric series is:

$S_n = u_1 \cdot \frac{r^n - 1}{r - 1}, \quad r \neq 1$

Both formulas appear in your formula booklet. When finding $r$, simply divide any term by its preceding term: $r = \frac{u_{n+1}}{u_n}$.

Infinite Geometric Series (Convergent Series)

When $|r| < 1$, the terms of a geometric sequence get progressively smaller and the series converges to a finite sum:

$S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1$

This is a powerful result. For example, the repeating decimal $0.333\ldots$ can be written as the geometric series $\frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots$ with $u_1 = 0.3$ and $r = 0.1$, giving $S_\infty = \frac{0.3}{0.9} = \frac{1}{3}$.

If $|r| \geq 1$, the series diverges — the sum grows without bound (or oscillates) and $S_\infty$ does not exist.

Sigma Notation

Sigma notation provides a compact way to write series. The expression:

$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n$

means "sum the terms $a_k$ as $k$ goes from 1 to $n$." For example:

$\sum_{k=1}^{5} (2k + 1) = 3 + 5 + 7 + 9 + 11 = 35$

On your GDC, you can evaluate sigma notation directly using the summation function, which is useful for checking answers.

Compound Interest and Real-World Applications

Geometric sequences model compound interest. If you invest a principal $P$ at an annual rate of $r\%$ compounded annually, the amount after $n$ years is:

$A = P\left(1 + \frac{r}{100}\right)^n$

This is simply a geometric sequence with first term $P$ and common ratio $\left(1 + \frac{r}{100}\right)$. Similarly, depreciation (losing a fixed percentage each year) is modelled by a geometric sequence with $r < 1$.

Arithmetic sequences model situations with constant change, such as a salary that increases by a fixed amount each year or a swimming pool being filled at a constant rate.

Tip 1: Identify the Sequence Type First

Before applying any formula, determine whether you are dealing with an arithmetic or geometric sequence. Check: is the difference between consecutive terms constant (arithmetic), or is the ratio constant (geometric)? This single step prevents most errors.

Tip 2: Use Simultaneous Equations

If given two terms (e.g., $u_3 = 12$ and $u_7 = 28$), set up two equations using the general term formula and solve simultaneously. For arithmetic: $u_3 = u_1 + 2d$ and $u_7 = u_1 + 6d$. Subtract to find $d$, then back-substitute for $u_1$.

Tip 3: Check Convergence Before Using $S_\infty$

The infinite sum formula only works when $|r| < 1$. Always verify this condition. If the question asks whether a series converges, you must explicitly state why $|r| < 1$ (or why it does not).

Tip 4: Use Your GDC Strategically

Your GDC can generate sequences, evaluate sums, and solve equations. Use the table function to list terms and verify patterns. For sigma notation problems, the GDC's summation feature gives instant confirmation of your hand-calculated answer.

Tip 5: Relate Back to Context

IB questions often embed sequences in real-world contexts. After solving, always interpret your answer in context and check it makes sense (e.g., a negative number of people is impossible).