Question

an AP consists of 37 terms the sum of the three middle most term is 225 and the sum of the last three terms is 429 find the AP

Answer 1

Three middle terms are 18,19,20th terms
a+17d+a+18d+a+19d=225
3a+54d=225
a+18d=75

a+34d+a+35d+a+36d=429
3a+105d=429
a+35d=143

By solving equations a=3 and d=4
AP is 3,7,11,15,.....,147

Answer 2

Let the first term and the common difference of the A.P are a and d respectively.
Since the A.P contains 37 terms. So, the middle most term is (37+1)/2 th term = 19th term.
Thus, three middle most terms of this A.P.are 18th, 19th and 20th terms.
Given a18 + a19 + a20 = 225
⇒ (a + 17d) + (a + 18d) + (a + 19d) = 225
⇒ 3(a + 18d) = 225
⇒ a + 18d = 75
⇒ a = 75 – 18d … (1)

According to given information
a35 + a36 + a37 = 429
⇒ (a + 34d) + (a + 35d) + (a + 36d) = 429
⇒ 3(a + 35d) = 429
⇒ (75 – 18d) + 35d = 143
⇒ 17d = 143 – 75 = 68
⇒ d = 4

Substituting the value of d in equation (1), it is obtained
a = 75 – 18 × 4 = 3

Thus, the A.P. is 3, 7, 11, 15 …

Hope it helps u....!!!