Understanding the Sum of a Geometric Sequence: A Practical Guide

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This article will guide you through the mathematical steps to find the sum of the first n terms of a geometric sequence, specifically for the sequence 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049.

Identifying the Sequence and its Parameters

The given sequence 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049 is a geometric sequence with the first term a 3 and common ratio r 3. The common ratio is found by dividing any term by the preceding term, for example, r frac{9}{3} 3.

Summation Formula for a Geometric Sequence

The formula for the sum of the first n terms of a geometric sequence is given by:

S_n a frac{1 - r^n}{1 - r} quad text{for } r eq 1

Where: - a is the first term - r is the common ratio - n is the number of terms

Calculating the Sum of the First 6 Terms

Let's calculate the sum of the first 6 terms of the given sequence.
First term, a 3
Common ratio, r 3
Number of terms, n 6

Substitute the values into the formula:

S_6 3 frac{1 - 3^6}{1 - 3} 3 frac{1 - 729}{-2} 3 times 364 1092

Thus, the sum of the first 6 terms of the geometric sequence is 1092.

Brute Force Solutions

A brute force solution involves generating the sequence and summing its elements directly.

Using the J Programming Language

To generate the sequence and sum its elements in the J programming language:

Generate the sequence: 3^1 2 3 4 5 6
Output: (3 9 27 81 243 729)
Sum the elements: / 3^1 2 3 4 5 6
Output: 1092

General Form and Equation Solving

For the sequence where the first term is 3 to the power of 1 and the 6th term is 3 to the power of 6, the common ratio r can be found by solving the equations:

Tn ar^n-1

For the second term: 81r^2 - 1 27
r frac{9}{27} frac{1}{3}

Note that the sum formula can then be applied as:

S_n a frac{1 - r^n}{1 - r} 81 frac{1 - 1/3^6}{1 - 1/3} 1092

thus, the sum of the first 6 terms is 1092.