Final answer:
To find the sum of the series s = 3/1 + 9/1 + 27/1 + ... + (3/1)ⁿ + ..., we can use the binomial theorem and the formula for the sum of a geometric series.
Step-by-step explanation:
To find the sum of the series s = 3/1 + 9/1 + 27/1 + ... + (3/1)ⁿ + ..., we can observe that each term is the cube of a power of 3. Let's rewrite the series using the binomial theorem: s = (3/1)³ + (3/1)⁴ + (3/1)⁵ + ... + (3/1)ⁿ. Now, we can use the formula for the sum of a geometric series, where a is the first term and r is the common ratio: S = a * (1 - rⁿ) / (1 - r). In this case, a = 3, r = 3/1, and n is the number of terms. Plugging these values into the formula, we get S = 3 * (1 - (3/1)ⁿ) / (1 - 3/1). Simplifying further, we have S = 3 * (1 - (3/1)ⁿ) / (-2). Therefore, the sum of the given series is S = (3/2) * ((3/1)ⁿ - 1).