Sequence And Series

To solve the problem, we need to identify the pattern in the given series and determine a formula for the sum of its first terms.

Observe the numerators: 4, 10, 28, ...

These numerators can be observed as terms derived from the sequence

Generalizing, the numerator for the -th term is .

The denominators are powers of 3: 3, 9, 27, ..., which can be written as or in general as .

The -th term is .

Let's derive the formula for the sum of the series until the nth term:

The series can be expressed as:

Splitting into two separate sums, we have:

Simplifying each part, the first part becomes a geometric series:

This sum is:

For the second part:

This is a geometric series with the first term and a common ratio :

Combining both sums:

This simplifies to the correct given answer:

Thus, the sum of the series is .