1.11 Sum of infinite geometric sequencesIB Maths: Applications and Interpretation HL: Revision notes

A geometric sequence has a constant ratio $r$ between terms: $u_n=u_1r^{n-1}$. The sum of the first $n$ terms is $S_n=\frac{u_1\left(1-r^{n}\right)}{1-r},\qquad r\ne1.$ Example: $u_1=12$, $r=\frac13$ gives $u_4=12\left(\frac13\right)^{3}=\frac49$ and $S_n=18\left(1-\left(\frac13\right)^{n}\right)$. These formulae are in the formula booklet.

If $|r|<1$ then $r^{n}\to0$ as $n\to\infty$, so the partial sums $S_n$ settle on a limit. The series is convergent and its sum to infinity is the limit of $S_n$: $S_\infty=\frac{u_1}{1-r},\qquad|r|<1.$ If $|r|\ge1$ the terms do not shrink, the series is divergent and there is no sum to infinity. For $u_1=12$ and $r=\frac13$: $S_\infty=\frac{12}{\frac23}=18$. The partial sums $S_5=17.93\ldots$ are already very close.

A recurring decimal is an infinite geometric series. For $0.272727\ldots=0.27+0.0027+0.000027+\ldots$ the first term is $u_1=0.27$ and the ratio is $r=0.01$, so $S_\infty=\frac{0.27}{1-0.01}=\frac{0.27}{0.99}=\frac{27}{99}=\frac{3}{11}.$ For $0.7777\ldots$: $u_1=0.7$, $r=0.1$, so $S_\infty=\frac{0.7}{0.9}=\frac79$. Group the repeating block to find $u_1$, and the ratio is $10^{-k}$ for a block of $k$ digits.

Many situations repeat a fixed percentage change. A ball dropped from $2$ m that rises to $80\%$ of its previous height has rebound heights $1.6,1.28,\ldots$ with $r=0.8$. The total distance is $2+2\times\frac{1.6}{1-0.8}=18$ m, because every rebound is travelled twice (up and down). For a daily drug dose of $200$ mg with $70\%$ remaining after each day, the amount just after the $n$th dose is $S_n=\frac{200\left(1-0.7^{n}\right)}{0.3}$. It approaches $\frac{200}{0.3}=667$ mg and never exceeds it. To find when a level is passed, solve $S_n>k$ with your GDC (table or logarithms).

Limit: $S_\infty$ is the limit of $S_n$. Fractals: in a self-similar shape such as a repeated triangle pattern, the perimeters or areas at each stage often form a geometric sequence, so the total can be found with $S_\infty$ when $|r|<1$. Markov chains: the long-term (steady) behaviour of a process that changes by fixed proportions each step is also a limit of repeated multiplication.