IB Mathematics Applications & Interpretation AHL 1.11—sum of Infinite Geometric Sequences Questionbank

(c) Show that the number of crates sent out in round $n$ is $5500(0.78)^{n-1}$.

(e) Explain why the total number of deliveries the scheme can ever make is finite, and find this total.

(f) Show that the total number of crate journeys the scheme generates, outward and return together, is $44500$.

One crate-kilometre is one crate carried one kilometre.

(g) Find the total number of crate-kilometres the scheme generates, and the number generated by the outward journeys alone.

A new crate costs $\text{＄}9.40$, and the depot buys the $5500$ crates of the first round at the start. Give all amounts of money correct to the nearest cent.

(j) Determine whether the depot should adopt this proposal.

For a geometric progression, the third term is $6$ and the sixth term is $\frac{2}{9}$.

(a) Determine the common ratio for this progression.

(b) Calculate the sum to infinity for this specific progression.

Two linked feedback controllers produce a combined signed displacement $3r^k-2(-r)^k$ millimetres at step $k$, starting with $k=0$. The theoretical total is

$S(r)=\sum_{k=0}^{\infty}\left(3r^k-2(-r)^k\right).$

The gain parameter $r$ is real.

A pump delivers successively smaller positive amounts of liquid, forming a geometric sequence. The model predicts a total of 45 litres after infinitely many deliveries. The first three deliveries total 42.12 litres.

(a) Determine the common ratio and the first term.

(b) Find the least positive integer $n$ for which the sum of the first $n$ terms is within $0.01$ of $45$.

A water purification system flushes contaminated water from a storage tank. In the first hour, $768$ litres are removed. As the pressure falls, the amount removed in each subsequent hour is $\frac34$ of the amount removed in the hour immediately before it.

(c) State one limitation of the mathematical model provided.

(d) Calculate the total number of crate deliveries made in the first $8$ rounds.

(h) Calculate the cost of crates per delivery under the scheme.

The depot's haulage contract is written for one batch of crates. It allows the depot a total of $1600000$ crate-kilometres, from the first delivery of the batch to the last return.

(i) Determine whether the scheme as it stands fits within this total, and find the largest return proportion, as a whole number of per cent, for which it does.

The transport manager proposes raising the return proportion to $85\ \%$. She says that this would cut the depot's crate cost per delivery by about a third and would barely change the total crate-kilometres.

(a) Determine the values of $r$ for which $S(r)$ exists, and obtain a closed expression for its value.

(b) Determine all values of $r$ for which $S(r)=3$.

(c) For the valid value from part (b), let $S_n$ be the sum of the terms with $0\le k\le n$. Derive the error $S(r)-S_n$ separately for even and odd $n$, and hence find the least $n$ for which this error is less than $0.001$.

(c) A display rounds the cumulative delivery to the nearest 0.1 litre, with halfway values rounded upwards. Find the first $n$ for which it displays 45.0 litres. Determine whether this reading justifies the tolerance achieved in part (b).

(a) Calculate the total volume of water removed during the first $6$ hours.

(b) The tank contains $3200$ litres of contaminated water. Determine whether the system will remove all of this water based on the model.