We know that,
First term of an AP = a
Common difference of AP = d
nth term of an AP, an = a + (n – 1)d
Since, n = 37 (odd),
Middle term will be (n+1)/2 = 19th term
Thus, the three middle most terms will be,
18th, 19th and 20th terms
According to the question,
a18 + a19 + a20 = 225
Using an = a + (n – 1)d
a + 17d + a + 18d + a + 19d = 225
3a + 54d = 225
3a = 225 – 54d
a = 75 – 18d … (1)
Now, we know that last three terms will be 35th, 36th and 37th terms.
According to the question,
a35 + a36 + a37 = 429
a + 34d + a + 35d + a + 36d = 429
3a + 105d = 429
a + 35d = 143
Substituting a = 75 – 18d from equation 1,
75 – 18d + 35d = 143 [ using eqn1]
17d = 68
d = 4
Then,
a = 75 – 18(4)
a = 3
Therefore, the AP is a, a + d, a + 2d….
i.e. 3, 7, 11….