Understanding the Sum of the First 5 Terms in a Geometric Sequence

The problem at hand involves understanding and calculating the sum of the first 5 terms of a given geometric sequence: 1, 3, 9, 27. This article will detail the steps to find the sum of these terms using a geometric sequence formula.

Identifying the Key Components of a Geometric Sequence

A geometric sequence is defined by its first term, denoted as (a), and a common ratio, denoted as (r). In this sequence, the first term is (a 1) and the common ratio (r 3), as each term is three times the previous term.

Using the Geometric Series Formula

The sum (S_n) of the first (n) terms of a geometric sequence can be calculated using the formula:

(S_n a frac{1 - r^n}{1 - r})

Applying the Formula to Our Sequence

Given (a 1), (r 3), and (n 5), we can substitute these values into the formula:

(S_5 1 frac{1 - 3^5}{1 - 3})

Step-by-Step Calculation

Calculate (3^5): (3^5 243) Substitute back into the formula: (S_5 1 frac{1 - 243}{1 - 3}) Simplify the expression: (S_5 1 frac{-242}{-2}) Result: (S_5 1 times 121 121)

Discussion on Irregular Sequences and Extensions

It is noteworthy that the sequence given extends indefinitely, with each term being a power of 3. However, in many practical applications, we may encounter finite sequences or sequences with varying patterns that do not follow a simple power progression.

For example, consider the sequence: 1, 3, 9, 27, 81, 243. To find the sum of the first six terms, we simply add these values together:

(1 3 9 27 81 243 364)

Other sequences might be divisors of 999, leading to different sums. For instance, the sequence 1, 3, 9, 27, 81, 243, 729 could be the divisors of some number and have a sum of 364. Another possible sequence is 1, 3, 9, 27, 81, 243, which has a sum of 364.

Brute Force Solution

For educational purposes and to ensure accuracy, it’s useful to implement a brute force approach using a programming language like J. The following steps are provided:

Generate the first 6 terms of the sequence using powers of 3. Add these terms together.

Using J programming language:

3^i.6 1 3 9 27 81 243

Adding the terms:

3^i.6 364

This confirms the sum of the first 6 terms is 364.