Sum of Geometric Progression

To solve this problem, we need to find the sum of a geometric progression. A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a fixed number. The fixed number is called the common ratio. In this case, the common ratio is 1/3.

Formula for Sum of Geometric Progression

The formula for the sum of a geometric progression is:

S = a(1 - r^n) / (1 - r)

Where:
- S is the sum of the geometric progression
- a is the first term of the geometric progression
- r is the common ratio of the geometric progression
- n is the number of terms in the geometric progression

Applying the Formula

In this problem, we are given the first term a = 1 1/3, the common ratio r = 1/3, and the number of terms n = 4. We can substitute these values into the formula and simplify.

S = (4/3)(1 - (1/3)^4) / (1 - 1/3)
S = (4/3)(1 - 1/81) / (2/3)
S = (4/3)(80/81) / (2/3)
S = (320/243) / (2/3)
S = (320/243) * (3/2)
S = 160/81

Therefore, the sum of 1 1/3, 1/3², 1/33, ..., 1/3ⁿ⁻¹ is 160/81.