Let's solve this arithmetic progression (AP) problem step by step.
We are given that the sum of the first 18 terms of the AP is 45. We can use the formula for the sum of an AP:
Sn = n/2 * (2a + (n - 1) * d)
where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Substituting the given values, we have:
45 = 18/2 * (2a + (18 - 1) * d)
45 = 9 * (2a + 17d)
45 = 18a + 153d ----(1)
We are also given that the 2nd term is twice the 17th term. Let's use this information to form another equation.
a + d = 2(a + 16d)
a + d = 2a + 32d
a - 2a = 32d - d
-a = 31d ----(2)
Now, we have a system of equations (equation 1 and equation 2) that we can solve to find the first term (a) and the common difference (d).
Multiplying equation (2) by 18, we get:
-18a = 558d ----(3)
Now, we subtract equation (3) from equation (1):
45 - (-18a) = 18a + 153d - 558d
45 + 18a = 18a - 405d
18a - 18a = -405d - 45
0 = -405d - 45
405d = -45
d = -45 / 405
d = -1/9
Substituting the value of d into equation (2):
-a = 31d
-a = 31 * (-1/9)
-a = -31/9
a = 31/9
Therefore, the first term (a) of the AP is 31/9 and the common difference (d) is -1/9.