for the sequence, describe the pattern, write the next term, and write a rule for the nth term 1.) 2,4,8,16 2.)1,8,27,64 3.)1/1,1/4,1/9,1/16 4.)4/3,5/3,6/3,7/3 5.)3,5,7,9

1. Powers of 2 ([tex]2^1=2,\,2^2=4,\,2^3=8,\,2^4=16,\ldots [/tex]).

Next term = [tex]2^5=\boxed{32}[/tex]

[tex]\boxed{a_n=2^n}[/tex]

2. Cube numbers ([tex]1^3=1,\,2^3=8,\,3^3=27,\,4^3=64,\ldots[/tex])

Next term = [tex]5^3=\boxed{125} [/tex]

[tex]\boxed{a_n=n^3}[/tex]

3. Reciprocals of the square numbers ([tex]\dfrac{1}{1^2}=\dfrac{1}{1},\,\dfrac{1}{2^2}=\dfrac{1}{4},\,\dfrac{1}{3^2}=\dfrac{1}{9},\,\dfrac{1}{4^2}=\dfrac{1}{16},\ldots[/tex]

Next term = [tex]\dfrac{1}{5^2}=\boxed{\dfrac{1}{25}}[/tex]

[tex]\boxed{a_n=\frac{1}{n^2}}[/tex]

4. Numbers 4, 5, 6, 7,... divided by 3.

Next term = [tex]\dfrac{8}{3}[/tex]

[tex]\boxed{a_n=\dfrac{n+3}{3}}[/tex]

5. Odd numbers from 3.

Next term = 11

[tex]\boxed{a_n=2n+1}[/tex]