Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=1, r=2, n=12 $$

Step-by-Step Solution

Answer

The sum of the first 12 terms of the series is 4095.

Step 1: Identify the Formula

To find the sum of the first \( n \) terms of a geometric series, we use the formula: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \] where \( S_n \) is the sum, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 2: Substitute Known Values

Substitute the known values into the formula: \( a_1 = 1 \), \( r = 2 \), and \( n = 12 \). The formula becomes: \[ S_{12} = 1 \frac{2^{12} - 1}{2 - 1} \]

Step 3: Simplify the Formula

Since \( 2 - 1 = 1 \), the formula simplifies to: \[ S_{12} = 2^{12} - 1 \]

Step 4: Calculate \(2^{12}\)

Calculate \( 2^{12} \): \[ 2^{12} = 4096 \].

Step 5: Calculate the Final Result

Subtract 1 from 4096 to find the sum: \[ S_{12} = 4096 - 1 = 4095 \].

Key Concepts

Understanding the Sum of Series

The sum of a series is the total obtained when all terms of the series are added together. In the context of a geometric series, this involves summing up all the terms generated by a specific rule. A geometric series 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. The sum of the first \( n \) terms of a geometric series can be calculated using a specific formula:

• \( S_n = a_1 \frac{r^n - 1}{r - 1} \) where \( S_n \) represents the sum of the first \( n \) terms,
• \( a_1 \) is the first term of the series,
• \( r \) is the common ratio between consecutive terms,
• \( n \) is the number of terms.

This formula is derived by manipulating the terms of the series and utilizing the properties of exponents. The beauty of this formula is that it allows us to find the sum without having to manually add each term, which is incredibly useful for large numbers of terms.

Geometric Progression: What It Means

A geometric progression (GP) refers to a sequence of numbers where each term after the first is found by multiplying the previous term by a consistent factor known as the common ratio. This makes it distinct from an arithmetic progression, where terms increase by a constant difference.
In our problem, the first term \( a_1 \) was 1, and the common ratio \( r \) was 2, forming the sequence:

• 1, 2, 4, 8, 16, 32, ...

This series highlights how quickly terms in a geometric progression can grow, especially with a common ratio greater than 1. Such sequences are prevalent in many real-world scenarios, including population growth and financial investments, due to their exponential nature. Understanding the behavior of geometric progressions is crucial for exploring a variety of applications.

Unpacking the Sequence Formula

The sequence formula for a geometric progression is an essential tool to find any term in the sequence. This formula is expressed as:

• \( a_n = a_1 \cdot r^{(n-1)} \)

Here, \( a_n \) is the \( n^{th} \) term you want to find, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.
Using the formula, you can efficiently calculate any term without having to compute all preceding terms. For example, if you need to find the 5th term in a sequence where \( a_1 = 1 \) and \( r = 2 \), you simply apply:

• \( a_5 = 1 \cdot 2^{4} = 16 \)

This formula simplifies working with long sequences and ensures accuracy when solving problems involving geometric progressions.