SOLUTION: Can the difference of any two terms of the arithmetic sequence 10,17,24,...be 100? Justify your answer.

10,17,24,...

Let's find out if this arithmetic sequence can have two terms
that differ by 100..  The formula for the nth term of an
arithmetic sequence is

an = a1 + (n-1)d

The common difference d = 17-10 = 24-17 = 7 and the first term
a1 = 10.  Substituting:

an = 10 + (n-1)(7)
an = 10 + 7(n-1)
an = 10 + 7n - 7
an = 3 + 7n

Let's find out whether the pth and qth terms 
can differ by 100 for some p and q, where p > q and p and q 
are positive integers.

ap = 3 + 7p
aq = 3 + 7q

Set their difference ≟ 100.  (The question mark over the =
indicates that we are not sure that they can be equal.)

 (3 + 7p) - (3 + 7q) ≟ 100

     3 + 7p - 3 - 7q ≟ 100

             7p - 7q ≟ 100

            7(p - q) ≟ 100

               p - q ≟ 

 is not an integer.  However p and q
are integers, and the difference of two integers 
is always an integer.  So p - q cannot equal 

So the answer to the question asked is no.
    
Edwin