IB Maths AI 1.3 Definitions

The constant multiplier between consecutive terms of a geometric sequence, found using $r=\frac{u_{n+1}}{u_n}$.

Approaching a finite limit; an infinite geometric series is convergent only when $|r|\lt1$.

Totals found by adding only a fixed number of terms from a series; for a geometric series this is written $S_n$ and can be calculated using $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$ when $r\neq1$.

A sequence in which each term is obtained by multiplying the previous term by a constant value called the common ratio.

A series formed by adding the terms of a geometric sequence, for example $u_1+u_1r+u_1r^2+\dots$.

A formula for the general term of a sequence; for a geometric sequence it is given by $u_n=u_1r^{n-1}$.

The notation for the total of the first $n$ terms of a series; for a geometric series with $r\neq1$, $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$, and if $r=1$ then $S_n=nu_1$.

A compact way to write a sum using the symbol $\sum$, for example $\sum_{k=0}^{n-1}u_1r^k$ for the first $n$ terms of a geometric series.

The total of the first $n$ terms of a series, written $S_n$; for a geometric series with $r\neq1$, $S_n=\frac{u_1\left(1-r^n\right)}{1-r}$.

The finite value approached by an infinite geometric series when it converges; if $|r|\lt1$ then $S_\infty=\frac{u_1}{1-r}$.