Problem 39 - Algebra and Trigonometry

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a=5, \quad r=2, \quad n=6 $$

Step-by-Step Solution

Answer

The partial sum \( S_6 \) is 315.

Step 1: Identify the Formula

To find the partial sum of a geometric sequence, we need to use the formula for the sum of the first n terms of a geometric sequence: \( S_n = a \frac{r^n - 1}{r - 1} \). Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 2: Substitute the Given Values

Substitute the given values into the formula: \( a = 5 \), \( r = 2 \), and \( n = 6 \). Thus, the formula becomes: \( S_6 = 5 \frac{2^6 - 1}{2 - 1} \).

Step 3: Calculate \( r^n \)

Calculate \( 2^6 \). Since \( 2^6 = 64 \), plug this value back into the formula: \( S_6 = 5 \frac{64 - 1}{2 - 1} \).

Step 4: Simplify the Formula

Simplify the terms inside the fraction: \( 64 - 1 = 63 \). Thus, the formula simplifies to \( S_6 = 5 \frac{63}{1} \).

Step 5: Final Calculation

Since \( \frac{63}{1} = 63 \), the formula becomes \( S_6 = 5 \times 63 \). Calculate this multiplication to get \( S_6 = 315 \).

Key Concepts

Partial Sum

The partial sum is a crucial concept in understanding sequences. In the case of a geometric sequence, the partial sum refers to the sum of a finite number of terms, denoted typically as \( S_n \). This sum provides insight into the behavior of the sequence up to the \( n \)-th term.
When working with geometric sequences, the formula for calculating the partial sum is:

• \( S_n = a \frac{r^n - 1}{r - 1} \)

In this formula:

• \( a \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the number of terms you wish to sum.

Understanding the partial sum allows you to quickly calculate the total of selected terms in the sequence, providing a useful tool for solving more complex mathematical problems.

Common Ratio

The common ratio, denoted as \( r \), is a fundamental component of a geometric sequence. It describes the consistent factor by which each term in the sequence is multiplied to get the next term.
In the provided exercise, the common ratio is 2, indicating each term is twice the previous one. Here's how you can identify and use the common ratio:

• Look at any two consecutive terms in the sequence.
• Divide the second term by the first. For example, if the terms are 5 and 10, the ratio is \( 10/5 = 2 \).

This value of \( r \) allows us to construct the entire sequence once the first term is known. This property makes the common ratio extremely valuable, especially when working with evolving series where predicting future terms can be essential.

First Term

In any sequence, the first term, denoted as \( a \), serves as the starting point. It is the foundation upon which all subsequent elements of the sequence are built.
In the geometric sequence from the exercise, the first term \( a \) is given as 5. Knowing this first term is vital because:

• It establishes the beginning of the sequence.
• It is prominently featured in the formula for calculating the partial sum, influencing all future sums.
• It determines the scale of the sequence, influencing the magnitude of the terms.

Understanding and identifying the first term helps in setting the correct sequence and ensures all calculations based on the sequence are accurate.