Arithmetic Progression Example

                                               Arithmetic Progression Example

Q: The sums of n terms of two arithmetic progressions are in the ratio 5n+4:9n+6. Find the ratio of their 18^th terms.

                                                                   Solution.

Let a1,a2  and d1,d2 be the first terms and common differences of the first and the second arithmetic progression respectively. Now, as per the problem:

                                              Sum of the n terms of the first AP                            5n + 4

                                          --------------------------------------------------     =      -------------------   

                                            Sum of the n terms of the second AP                           9n+16

or,

                  n/2[2a1+ (n-1)d1]                                            5n+4

           -----------------------------------      =             ---------------------------------

                    n/2[2a2+ (n-1)d2]                                          9n+16

or,    

                 [2a1+ (n-1)d1]                                         5n+4   

     -----------------------------------   =             ------------------------------              ……..(1)

                 [2a2+ (n-1)d2]                                         9n+16    

                  Now,                           

              18^th Term of first AP                                     a1+17d1   

     -----------------------------------          =        --------------------------------     ………(2)

             18^th Term of second AP                                 a2+17d2

              Multiplying Numerator and Denominator of  (2) by  2, we have

                                        2a1+34d1                   5 * 35  + 4

                                 -------------------------- =  ----------------

                                        2a2+34d2                    9 * 35 + 6

               Therefore, we have,

           a2+17d2                                                    179

      --------------------------   =       ----------------------------------------------

           a2+17d2                                                      321

Thus, the required ratio is 179 / 321.