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Arithmetic Progression Example

Ashish K Sharma
23/07/2022 0 0

                                               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 18th 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,                           

              18th Term of first AP                                     a1+17d1   

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

             18th 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.

 

 

 

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