(i)   "the sum of the fourth and ninth terms is 58"  means

       = 58,    or

       = 58,   or

       = 58.     (1)



(ii)  "the sum of the first 26 terms is 390"  means  that

       =  = 15,

      since the sum of the first n terms of any AP is the average of the first and n-th terms taken n times.


      The last equation means that

       = 30,   or

       = 30.      (2)



Now subtract eq(1) from eq(2).  You will get


       25d - 11d = 30 - 58,   or

       14d       = -28

which implies  d = -2.


Then from eq(1) you have   = 58 - 11*(-2) = 58 + 22 = 80.


Thus the AP has   = 80  and  d= -2.    ANSWER to question (a)


Now it is obvious that the first 39 terms of the AP are uniformly decreasing from 80 to 2 with the step -2;

the 40-th term is 0 (zero);

next 39 terms from    to    are negative from -2 to -80 and the sum of the first 79 terms is equal to zero.


The smallest integer value of "n" for which the sum to n terms is negative is 80.    ANSWER to question (b).