An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.


Let the a be first term and d be common difference of an AP


As n = 37 (odd)


Middle term will be



Three middle most terms will be


18th, 19th and 20th terms


Given,


a18 + a19 + a20 = 225


a + 17d + a + 18d + a + 19d = 225 [ using an = a + (n - 1)d]


3a + 54d = 225


3a = 225 - 54d


a = 75 - 18d [ eqn 1]


Also last three terms will be 35th , 36th and 37th terms


Given,
a35 + a36 + a37 = 429


a + 34d + a + 35d + a + 36d = 429


3a + 105d = 429


a + 35d = 143


75 - 18d + 35d = 143 [ using eqn1]


17d = 68


d = 4


using this value in eqn 1


a = 75 - 18(4)


a = 3


so AP is a, a + d, a + 2d….


i.e. 3, 7, 11….


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