Introduction:
We are given an arithmetic progression (AP) with the first term a1 = 1 and the third term a3 = 3a8. We need to find the largest possible value of the sum of the first n terms, denoted as sn.

Given:
a1 = 1
a3 = 3a8

Approach:
To find the largest possible value of sn, we need to determine the common difference (d) and the number of terms (n) in the arithmetic progression. Once we have these values, we can use the formula for the sum of an arithmetic series to calculate sn.

Finding the Common Difference:
We are given that a3 = 3a8. In an arithmetic progression, the nth term can be expressed as an = a1 + (n-1)d, where d is the common difference. Substituting the given values, we have:
a3 = a1 + (3-1)d
3a8 = 1 + 2d

We can solve these two equations to find the value of d.

Finding the Number of Terms:
To find the value of n, we need to identify the relationship between a1 and a8. Using the formula for the nth term, we have:
a8 = a1 + (8-1)d
Substituting the value of a1 and d from the previous step, we get:
a8 = 1 + 7d

Solving for d:
Equating the two expressions for a8, we have:
1 + 7d = 3a8
1 + 7d = 3(1 + 7d)
1 + 7d = 3 + 21d
14d = 2
d = 1/7

Solving for n:
Using the expression for a8, we have:
a8 = 1 + 7d
a8 = 1 + 7(1/7)
a8 = 2

Since a8 represents the 8th term, the number of terms (n) is 8.

Calculating sn:
Now that we have the values of d and n, we can calculate sn using the formula for the sum of an arithmetic series:
sn = (n/2)(a1 + an)

Substituting the values, we have:
sn = (8/2)(1 + 2)
sn = 4(3)
sn = 12

Thus, the largest possible value of sn is 12.