Review practice · Section 3 of 4

The sequence satisfies $x_{n+1}=12/(7-x_n)$ for positive integers $n$.

(i) Find the exact value(s) of $s$, the limit of $x_n$.3 marks

(ii) Show that if $3<x_n<4$, then $x_{n+1}<x_n$.3 marks

An insect colony initially has $50$ insects. Each week its population doubles, then $30$ die through competition and limited resources. Let $u_n$ be the population at the beginning of week $n$, $n=1,2,\ldots$, with all remaining insects alive.

(i) Form a recurrence relating week n and week n−1.1 mark

(ii) Find the minimum number of weeks needed for the population to exceed $300000$.2 marks

The cubic $x^3-6x+1=0$ has a root $\alpha$ in $(0,1)$.

(i) Find $\alpha$ to 5 decimal places.1 mark

(ii) For $x_{n+1}=(x_n^3+1)/6$, $x_1=0$, use a calculator to determine the sequence’s behaviour.2 marks

(iii) Prove algebraically that, if the sequence converges to $\alpha$, its limit is the root of the given cubic.2 marks

A storage tank loses $10\%$ of its water each day, then receives $90$ litres at the end of the day. Let $x_n$ be the volume after n days and $x_0$ its initial volume.

(i) Give a recurrence.1 mark

(ii) Show $x_n=0.9^n(x_0-900)+900$.3 marks

(iii) Deduce the limiting volume.1 mark

The second, fifth and tenth terms of an arithmetic progression with nonzero common difference are the first three terms of a geometric progression.

(i) Find the common ratio.3 marks

(ii) The first terms of the AP and GP are $7$ and $9$. Find the least n for which the GP’s n-th term exceeds the AP’s n-th term by at least $1000$.3 marks

(a) A sequence has partial sums $S_n=n^2+6n$.

(i) Find and simplify $u_n$.3 marks

(ii) Show it is an AP.2 marks

(iii) Find $u_1+u_3+\cdots+u_{49}$.2 marks

(b) A GP has nonzero first term a and nonzero ratio r with $-1<r<1$. Its (n−1)-th term equals the sum after, excluding, its n-th term.

(i) Show $r^2+r-1=0$.3 marks

(ii) Deduce r.1 mark