Question 256103
Lets assume the sequence 2,5,8,11,... is an arithmetic sequence. The general form of the arithmetic sequence is


{{{a[n]=d*n+a[1]}}} where {{{a[n]}}} is the nth term, d is the difference, and {{{a[1]}}} is the first term


So lets find the difference between 2 terms (i.e. the difference between two terms)

====================================================================================================================


To find the difference, simply pick any term and subtract the previous term from that selected term


{{{5-2=3}}} Select the 2nd term (which is 5) and subtract the 1st term (which is 2) from it.


So we get a difference of {{{3}}}



Lets pick another pair of terms to verify:


{{{8-5=3}}} Select the 3rd term (which is 8) and subtract the 2nd term (which is 5) from it.


And again, we get a difference of {{{3}}}

-----------------------------------------------------


Lets pick another pair of terms to verify:


{{{11-8=3}}} Select the 4th term (which is 11) and subtract the 3rd term (which is 8) from it.


And again, we get a difference of {{{3}}}




====================================================================================================================

Since we've tested every consecutive pair of terms, we've verified that the sequence has a constant difference of  {{{3}}}. This means the sequence is arithmetic


Since the difference is {{{d=3}}} and the first term is {{{a[1]=2}}}, this means the arithmetic sequence is


{{{a[n]=3n+2}}}  where {{{n}}} starts at {{{n=0}}}



So the list of numbers 2,5,8,11... can be generated by the sequence


{{{a[n]=3n+2}}} where {{{n}}} starts at {{{n=0}}}


Since n=0 generates the first term, this means that the 23rd term is generated when n=22



{{{a[n]=3n+2}}} Start with the given formula.



{{{a[22]=3(22)+2}}} Plug in {{{n=22}}}



{{{a[22]=66+2}}} Multiply



{{{a[22]=68}}} Add



So the 23rd term is 68