Question

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

Answer 1

1. Powers of 2 ($2^1=2,\,2^2=4,\,2^3=8,\,2^4=16,\ldots$).

Next term = $2^5=\boxed{32}$

$\boxed{a_n=2^n}$

2. Cube numbers ($1^3=1,\,2^3=8,\,3^3=27,\,4^3=64,\ldots$)

Next term = $5^3=\boxed{125}$

$\boxed{a_n=n^3}$

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

Next term = $\dfrac{1}{5^2}=\boxed{\dfrac{1}{25}}$

$\boxed{a_n=\frac{1}{n^2}}$

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

Next term = $\dfrac{8}{3}$

$\boxed{a_n=\dfrac{n+3}{3}}$

5. Odd numbers from 3.

Next term = 11

$\boxed{a_n=2n+1}$