5.1 Sequences  (Page 9/25)

$a_n=1+{\left(\mathrm{-1}\right)}^n$ for $n\ge 1$

$a_n=n^2-1$ for $n\ge 1$

$a_1=1$ and $a_n=a_{n-1}+n$ for $n\ge 2$

$a_1=1,$ $a_2=1$ and $a_{n+2}=a_n+a_{n+1}$ for $n\ge 1$

Find an explicit formula for $a_n$ where $a_1=1$ and $a_n=a_{n-1}+n$ for $n\ge 2.$

Find a formula $a_n$ for the $n\text{th}$ term of the arithmetic sequence whose first term is $a_1=1$ such that $a_{n-1}-a_n=17$ for $n\ge 1.$

Find a formula $a_n$ for the $n\text{th}$ term of the arithmetic sequence whose first term is $a_1=\mathrm{-3}$ such that $a_{n-1}-a_n=4$ for $n\ge 1.$

Find a formula $a_n$ for the $n\text{th}$ term of the geometric sequence whose first term is $a_1=1$ such that $\frac{a_{n+1}}{a_n}=10$ for $n\ge 1.$

Find a formula $a_n$ for the $n\text{th}$ term of the geometric sequence whose first term is $a_1=3$ such that $\frac{a_{n+1}}{a_n}=1\text{/}10$ for $n\ge 1.$

Find an explicit formula for the $n\text{th}$ term of the sequence whose first several terms are $\left\{0,3,8,15,24,35,48,63,80,99\text{,…}\right\}.$ ( Hint: First add one to each term.)

Find an explicit formula for the $n\text{th}$ term of the sequence satisfying $a_1=0$ and $a_n=2a_{n-1}+1$ for $n\ge 2.$

$\left\{1,0,\mathrm{-1},0,1,0,\mathrm{-1},0\text{,…}\right\}$ ( Hint: Find where $\text{sin}x$ takes these values)

$\left\{1,\text{−}1\text{/}3,1\text{/}5,\text{−}1\text{/}7\text{,…}\right\}$

$a_1=1$ and $a_{n+1}=\text{−}{a}_n$ for $n\ge 1$

$a_1=2$ and $a_{n+1}=2a_n$ for $n\ge 1$

$a_1=1$ and $a_{n+1}=\left(n+1\right)a_n$ for $n\ge 1$

$a_1=2$ and $a_{n+1}=\left(n+1\right)a_n\text{/}2$ for $n\ge 1$

$a_1=1$ and $a_{n+1}=a_n\text{/}{2}^n$ for $n\ge 1$

[T] $a_1=1,$ $a_2=2,$ and for $n\ge 2,$ $a_n=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right);$ $N=30$

[T] $a_1=1,$ $a_2=2,$ $a_3=3$ and for $n\ge 4,$ $a_n=\frac{1}{3}\left(a_{n-1}+a_{n-2}+a_{n-3}\right),$ $N=30$

[T] $a_1=1,$ $a_2=2,$ and for $n\ge 3,$ $a_n=\sqrt{a_{n-1}{a}_{n-2}};$ $N=30$

[T] $a_1=1,$ $a_2=2,$ $a_3=3,$ and for $n\ge 4,$ $a_n=\sqrt{a_{n-1}{a}_{n-2}{a}_{n-3}};$ $N=30$

$\underset{n\to \infty }{\text{lim}}3a_n-4b_n$

$\underset{n\to \infty }{\text{lim}}\frac{1}{2}{b}_n-\frac{1}{2}{a}_n$

$\underset{n\to \infty }{\text{lim}}\frac{a_n+b_n}{a_n-b_n}$

$\underset{n\to \infty }{\text{lim}}\frac{a_n-b_n}{a_n+b_n}$

$\frac{n^2}{2^n}$

$\frac{{\left(n-1\right)}^2}{{\left(n+1\right)}^2}$

$\frac{\sqrt{n}}{\sqrt{n+1}}$

$n^{1\text{/}n}$ ( Hint: $n^{1\text{/}n}=e^{\frac{1}{n}\text{ln}n})$