SOLUTION: An arithmetic series has the following properties (i) the sum of the fourth and ninth terms is 58. (ii) the sum of the first 26 terms is 390. (a) Find the first term and the com

(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).