Question

trying to figure out if the sequence is arithmetic or not.

Answer 1

In order to find if the sequence

$a_n=\frac{1}{3+8n}$

let's calculate some terms of the sequence given using the formula above

$a_1=\frac{1}{3+8(1)}=\frac{1}{3+8}=\frac{1}{11}$$a_2=\frac{1}{3+8(2)}=\frac{1}{3+16}=\frac{1}{19}$$a_3=\frac{1}{3+8(3)}=\frac{1}{3+24}=\frac{1}{27}$

The using the terms we found

If the difference between 1/19-1/11 is equal to 1/27-1/19 is equal therefore we have an arithmetic sequence

$a_2-a_1=\frac{1}{19}-\frac{1}{11}=-\frac{8}{209}$$a_3-a_2=\frac{1}{27}-\frac{1}{19}=-\frac{8}{513}$

As we can see they are different therefore the sequence is not arithmetic