An A.P. 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 A.P.

We are given the sum of the three middlemost terms of an A.P. and the sum of the last three terms, and we have to find the A.P.
The A.P. has 37 terms, so 𝑛 = 37.
Let the first term of the A.P. be 𝑎 and the common difference be 𝑑. The middlemost term of the A.P. is = 𝑛+1/2 
37+1/ 2
38/ 2
= 19
So, the 19𝑡ℎ term is the middlemost term.
The sum of the three middlemost terms = 225
The three middlemost terms are the 18𝑡ℎ, 19𝑡ℎ and 20𝑡ℎ terms. 

So, 𝑎₁₈ + 𝑎₁₉ + 𝑎₂₀ = 225
⇒ 𝑎 + (18 − 1)𝑑 + 𝑎 + (19 − 1)𝑑 + 𝑎 + (20 − 1)𝑑 = 225
⇒ 3𝑎 + 17𝑑 + 18𝑑 + 19𝑑 = 225
⇒ 3𝑎 + 54𝑑 = 225
⇒ 3(𝑎 + 18𝑑) = 225
⇒ 𝑎 + 18𝑑 = 225/3
⇒ 𝑎 + 18𝑑 = 75                               ....(1)
The sum of the last three terms = 429
The three last terms are the 35𝑡ℎ, 36𝑡ℎ and 37𝑡ℎ terms. 

So, 𝑎₃₅ + 𝑎₃₆ + 𝑎₃₇ = 429
⇒ 𝑎 + (35 − 1)𝑑 + 𝑎 + (36 − 1)𝑑 + 𝑎 + (37 − 1)𝑑 = 429
⇒ 3𝑎 + 34𝑑 + 35𝑑 + 36𝑑 = 429
⇒ 3𝑎 + 105𝑑 = 429
⇒ 3(𝑎 + 35𝑑) = 429
⇒ 𝑎 + 35𝑑 = 429/3 
⇒ 𝑎 + 35𝑑 = 143                               ....(2)
Subtracting eq(1) from eq(2), we get
𝑎 + 35𝑑 − (𝑎 + 18𝑑) = 143 − 75
⇒ 𝑎 + 35𝑑 − 𝑎 − 18𝑑 = 68
⇒ 17𝑑 = 68
⇒ 𝑑 = 68/17
⇒ 𝑑 = 4
So, the common difference between the A.P. is 4. Putting the value of 𝑑 in eq(1), we get
𝑎 + 18 × 4 = 75
⇒ 𝑎 + 72 = 75
⇒ 𝑎 = 75 − 72

⇒ 𝑎 = 3

So, the first term of the A.P. is 3.

Hence, the A.P. can be written as 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑,..., 𝑎 + 36𝑑

= 3, 7, 11, 15, ..., 147.