The last two equations give us the system



Simplify those equations and solve the system by substitution
or elimination.  You will then have the values for k and d. Then
find a1 by substituting the value for k in .

Then write out the first 17 terms beginning with a1 and 
add d over and over until you have 17 terms.

Oh, maybe by "17 term" you meant the "17th term" and not the "17 terms".

If so, use  with n=17. 

Do the other one the same way.

Edwin

Since three terms ,   and  form an arithmetic progression,
the difference   is the same as the difference  
(as each this difference is simply d, the common difference of the AP).


So, we write

    (5k-3) - (2k+1) = (7k-2) - (5k-3).


From this equation, we find the value of k

    5k - 3 - 2k - 1 = 7k - 2 - 5k + 3,

    3k - 4 = 2k +1

    3k - 2k = 1 + 4

        k   =   5.


Now we know the value of k and can restore the values of the first three twrms of this AP

     = 2*5+1 = 11;

     = 5*5-3 = 22;

     = 7*5-2 = 33.


Thus, the progression has first term 11 and the common difference d = 22-11 = 11.


At this point, we know everything about this AP and can easily find each of its terms.


In particular, the 17-th term of this AP is   =  + d*(n-1) = 11 + 11*(17-1) = 11 + 11*16 = 187.    ANSWER