Question

Find the sum of the first five terms of the geometric sequence with first term 3 and common ratio 2 .

Short answer

Question: Find the sum of the first five terms of a geometric sequence with a first term of 3 and a common ratio of 2. Answer: The sum of the first five terms of the geometric sequence is 93.

Step-by-step solution

Identify the first term and common ratio

The problem states that the first term (a_1) is 3 and the common ratio (r) is 2.

Use the formula for the sum of the first n terms of a geometric series

The formula for the sum of the first n terms of a geometric series is S_n = a_1 * (1 - r^n) / (1 - r). We are given a_1 = 3, r = 2, and n = 5, where n is the number of terms we want to include in the sum.

Plug the values into the formula and solve for S_n

Now, we plug in the given values into the formula: S_5 = 3 * (1 - 2^5) / (1 - 2)

Simplify the expression

The simplified expression is: S_5 = 3 * (1 - 32) / (-1)

Calculate the sum

S_5 = 3 * (-31) / (-1) = 93 So the sum of the first five terms of the geometric sequence with first term 3 and common ratio 2 is 93.

Key concepts

Geometric Series

A geometric series is a sequence 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. It’s akin to a chain reaction; once the initial term is set and the rate of multiplication is established, each subsequent term can propagate from its predecessor through consistent multiplication.

Applied to real-world scenarios, the principle of geometric series can be observed in contexts like population growth, where individuals reproduce at a constant rate, or in finance, when calculating compound interest. Understanding geometric series is fundamental to these and many other scientific and mathematical applications.

Common Ratio

The common ratio of a geometric series is the factor by which successive terms of the series are multiplied to obtain the next term. It's an essential component of the series' identity; altering it would change the progression of the series entirely. For a series written as 3, 6, 12, 24..., the common ratio is 2, as each term is double the previous one.

Determining the common ratio is straightforward when you have two consecutive terms—the next divided by the previous. A consistent common ratio leads to the mesmerizing feature of geometric series, which is their predictability and scalability.

Sum of N Terms Formula

Calculating the sum of the first 'n' terms in a geometric series involves a specific formula: \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \). Here, \( S_n \) represents the sum we’re seeking, \( a_1 \) is the first term, \( r \) the common ratio, and \( n \) the number of terms to sum.

This formula condenses the process into a simple substitution exercise. Rather than adding terms individually, which could be exceedingly tedious or outright impossible for large 'n', this formula provides a shortcut, a sum with minimal computational fuss. The utility of such a formula is inestimable, taking the grunt work out of finding sums and thereby fostering an efficiency in problem-solving.

Sequence and Series

In mathematics, the concepts of sequence and series are closely linked. A sequence is an ordered list of numbers following a particular rule, while a series is essentially the sum of the elements of a sequence. Think of a sequence as a collection of potential steps, and a series as the journey makes by stepping along that path, summing the distances covered.

The distinction is pivotal: sequences map out the potential, series tally the actual. In a sequence, you're noting each individual number; in a series, you're combining them into a cumulative total. Both concepts are fundamental to various fields within mathematics, including calculus, differential equations, and number theory, embodying the foundation upon which much of mathematical analysis rests.