Problem 41 - Algebra and Trigonometry

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{3}=28, \quad a_{6}=224, \quad n=6 $$

Step-by-Step Solution

Answer

The partial sum \( S_6 \) is 441.

Step 1: Understand the Geometric Sequence Formula

A geometric sequence can be expressed as \( a_n = a_1 imes r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. The formula for the partial sum \( S_n \) of the first \( n \) terms is \( S_n = a_1 \frac{1-r^n}{1-r} \) for \( r eq 1 \).

Step 2: Find Common Ratio

Given \( a_3 = 28 \) and \( a_6 = 224 \), we use the formula \( a_n = a_1 \, r^{n-1} \). Therefore: \( a_6 = a_3 \, r^{6-3} \). Substitute to find the common ratio: \( 224 = 28 \, r^3 \). Thus, \( r^3 = \frac{224}{28} = 8 \), and so \( r = 2 \).

Step 3: Find the First Term

Use \( a_3 = 28 \) and the formula \( a_3 = a_1 \, r^{3-1} \) to find \( a_1 \). Substitute for \( r \): \( 28 = a_1 \, 2^2 = a_1 \, 4 \), so \( a_1 = \frac{28}{4} = 7 \).

Step 4: Calculate the Partial Sum

Use the partial sum formula \( S_n = a_1 \frac{1-r^n}{1-r} \) with \( a_1 = 7 \), \( r = 2 \), and \( n = 6 \). Substitute these values: \( S_6 = 7 \frac{1-2^6}{1-2} = 7 \times \frac{1-64}{-1} = 7 \times 63 = 441 \).

Key Concepts

Partial Sum

The concept of a partial sum in a geometric sequence is crucial for understanding how individual terms add up to form a total. In a geometric sequence, terms increase or decrease by a constant factor called the common ratio. The partial sum, denoted as \( S_n \), refers to the sum of the first \( n \) terms of this sequence. Given a geometric sequence with a first term \( a_1 \) and a common ratio \( r \), the formula for the partial sum when \( r eq 1 \) is:
\[S_n = a_1 \frac{1-r^n}{1-r}\]
This equation allows you to quickly calculate the total sum of the initial terms without having to manually add each one. It is a powerful tool, especially when dealing with sequences that contain many terms. As with any formula, accuracy in identifying \( a_1 \), \( r \), and \( n \) is essential. By applying this formula, you can neatly encapsulate the cumulative effect of the sequence's growth pattern.

Common Ratio

The common ratio in a geometric sequence is the factor by which each term is multiplied to produce the next term. In our given exercise, we had two terms to work with: \( a_3 = 28 \) and \( a_6 = 224 \). By leveraging the relationship between these terms, we can solve for \( r \). Using the formula \( a_n = a_{n-k} \, r^k \) where \( k \) is the difference in term numbers, you find:

• \( a_6 = a_3 \, r^{6-3} = 224 \)
• \( 224 = 28 \, r^3 \)
• \( r^3 = 8 \rightarrow r = 2 \)

This calculation confirms that the common ratio is 2, meaning each term in the sequence is double the one before it. The common ratio is a key component because it dictates the rate at which the sequence grows or decays.

First Term

In a geometric sequence, the first term, denoted as \( a_1 \), serves as the starting point. It is essential because it scales the entire sequence. In this exercise, the first term needed to be derived using the known values for a particular term and the common ratio.
Starting with \( a_3 = 28 \) and knowing the common ratio \( r = 2 \), we use the formula for geometric sequences:

• \( a_3 = a_1 \, r^2 \)
• \( 28 = a_1 \, 4 \)
• \( a_1 = \frac{28}{4} = 7 \)

Here, the first term \( a_1 \) is determined to be 7. This value provides the baseline from which all other terms are generated by applying the common ratio. Correctly identifying \( a_1 \) allows for accurate computation of any other term using geometric sequence formulas.