Question

An arithmetic sequence starts as $5,9,13,..........$,. What is the next term? Is $2012$ a term of this sequence? Why?

Answer 1

Sequence is $5,9,13,............$
Common difference, $d=9-5=4$
Thus, next term $=13+4=17$
$t_n=a+(n-1)d$
$\Rightarrow t_n=5+(n-1)4=5+4n-4=1+4n$
That is, from any term of this sequence, if we subtract 1, then we get a multiple of 4.
Now, $2012-1=2011$
Since $2011$ is not a multiple of 4, the term 2012 is not a term of this sequence.