Practice section 1 of 17

(a) Write down the first four terms of (i) $u_n=1/n^2$, (ii) $u_n=2n-1$, for positive integers $n$. (b) Write down a general formula for (i) $1,1/2,1/3,1/4,1/5$; (ii) $3,5,7,9,11,\ldots$; (iii) $-1,1,-1,1,-1,\ldots$.

Describe the behaviour of the following sequences for $n\in\mathbb Z^+$: (a) $u_n=2n+1$; (b) $u_n=(-1/2)^n$; (c) $u_n=(-2)^n$; (d) $u_n=e^n$; (e) $u_n=2+e^{-n}$.

Without using a calculator, determine whether each sequence converges and, if it does, find the limit: (a) $u_n=\frac{2n+1}{(3n-1)^2}$; (b) $u_n=\frac{2n+1}{3n-1}$; (c) $u_n=\frac{6n^2+n+1}{3n-1}$.

Given $u_n=2u_{n-1}+3$, $u_1=2$, find $u_2,u_3,u_4$ and $u_{20}$.

Given $u_{n+1}-3u_n=4$, $u_1=5$, evaluate $u_2,u_3,u_4$.

The source AP table lists $2,7,12,17,22$; $13,10,7,4,1$; and $-14,-11,-8,-5,-2$. Complete the first-term $a$ and common-difference $d$ entries. The following printed question asks: is $2,-4,6,-8,10$ an arithmetic progression?