Practice section 15 of 17

Given $\sum_{r=1}^n1/(\sqrt{r+1}+\sqrt r)=\sqrt{n+1}-1$, deduce $\sum_{r=1}^n1/\sqrt r>2(\sqrt{n+1}-1)$. [2]

The n-th row from the top of a board has n tiles, labelled $1,2,\ldots,n$ from left to right, as in the diagram. Given $\sum_{r=1}^nr^2=n(n+1)(2n+1)/6$, find the sum in the k-th row and hence show the sum of all numbers in n rows is $n(n+1)(n+2)/6$. [4]

The sequence begins $1$, $(1+2)$, $(1+2+2^2)$, $(1+2+2^2+2^3)$, … .

(i) Show its n-th term is $2^n-1$. [2]

(ii) Find the sum of its first n terms. [3]

Find the first seven terms of $u_{n+1}=(n+1)u_n$, $u_1=1$, and a formula for $u_n$.

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]

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

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]

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