Revise Sequences and series: arithmetic and geometric for IB Mathematics: Analysis and Approaches (1.1)

An arithmetic sequence advances by ADDING a constant value, the common difference $d$, to reach each new term. You find it by subtracting any term from the one after it: $d = u_2 - u_1 = u_3 - u_2$. A positive $d$ gives an increasing sequence, a negative $d$ a decreasing one. Because you take equal-sized steps, the $n$th term is the first term plus $(n-1)$ of those steps.

\text{nth term: } \; u_n = u_1 + (n-1)d \[4pt] \text{Sum of first n terms: } \; S_n = \frac{n}{2}\big(2u_1 + (n-1)d\big) = \frac{n}{2}(u_1 + u_n)

A geometric sequence advances by MULTIPLYING by a constant value, the common ratio $r$, to reach each new term. You find it by dividing any term by the one before it: $r = u_2/u_1 = u_3/u_2$. This is the pattern behind anything that changes in proportion to its current size — money earning compound interest, a substance decaying, a population multiplying.

\text{nth term: } \; u_n = u_1 r^{\,n-1} \[4pt] \text{Sum of first n terms: } \; S_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}, \quad r \neq 1

Keep adding the terms of a geometric series forever and one of two things happens. If $|r| < 1$ each term is a fraction of the one before, the terms shrink towards zero, and the running total settles on a finite value — the series converges. If $|r| \ge 1$ the terms do not shrink and the total grows without bound — the series diverges. The convergent case has a compact formula for its limit.

$\text{Sum to infinity: } \; S_\infty = \frac{u_1}{1 - r}, \quad \text{valid only when } |r| < 1$.

Sigma notation is a compact way to write a sum. The expression $\displaystyle\sum_{k=1}^{n} u_k$ means 'add the terms $u_k$ as the index $k$ runs from $1$ up to $n$'. The letter below the sigma is the starting value of the index, the number above is the final value, and the formula to its right generates each term. A common exam task is to recognise an arithmetic or geometric series hiding inside sigma notation and then apply the right sum formula.

Any quantity that changes by a fixed percentage each period is geometric. A balance earning $r\%$ interest per year multiplies by $1 + \frac{r}{100}$ each year; a quantity decaying by $r\%$ per period multiplies by $1 - \frac{r}{100}$. So the value after $n$ periods is $u_1$ times that ratio to the power $n$. Note the subtle indexing: if $u_1$ is the amount at the START (time zero), the value after $n$ full periods is $u_1 r^{n}$, one power higher than the sequence's $n$th term — read the wording carefully to decide whether the initial amount counts as term 1 or as term 0.

In Paper 1 the marks are analytic: each one is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and follow-through (FT) means a wrong number early on need not cost you the marks that follow, provided your later method is correct and applied to your own figure. But that protection only exists if the method is written down. Study how each mark below is earned by a specific line in a classic 'find $u_1$ and $r$, then sum' question.

Sequences and series underpin far more of the syllabus than they first appear to. Geometric series reappear in financial mathematics and, at HL, in the binomial theorem and in proofs by induction; the idea of a convergent infinite sum previews limits and the foundations of calculus. Master the habit here — decide the type, extract $u_1$ and $d$ or $r$, pick the formula that omits the unknown you do not want, check $|r|<1$ before summing to infinity, and show every line — and the algebra that follows becomes variations on a method you already own.