Problem 45 - Precalculus Mathematics for Calculus

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

In a geometric sequence, each term is found by multiplying the previous term by a constant ratio \( r \). The \( n \)-th term of a geometric sequence can be expressed as \( a_{n} = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 2: Set equations for given terms

Based on the formula \( a_{n} = a_1 \cdot r^{n-1} \), we have two equations from the conditions: \( a_3 = a_1 \cdot r^2 = 28 \) and \( a_6 = a_1 \cdot r^5 = 224 \).

Step 3: Solve for the common ratio \( r \)

Divide the second equation by the first equation to eliminate \( a_1 \): \[ \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = \frac{224}{28} \implies r^3 = 8 \implies r = 2. \]

Step 4: Determine the first term \( a_1 \)

Substitute \( r = 2 \) into the equation \( a_1 \cdot r^2 = 28 \): \[ a_1 \cdot 2^2 = 28 \implies a_1 \cdot 4 = 28 \implies a_1 = 7. \]

Step 5: Calculate the partial sum \( S_6 \)

The formula for the sum \( S_n \) of the first \( n \) terms of a geometric sequence is: \[ S_n = a_1 \cdot \frac{r^n - 1}{r - 1}. \] Substituting \( a_1 = 7 \), \( r = 2 \), and \( n = 6 \): \[ S_6 = 7 \cdot \frac{2^6 - 1}{2 - 1} = 7 \cdot (64 - 1) = 7 \cdot 63 = 441. \]

Key Concepts

Partial Sum

A partial sum in a geometric sequence refers to the sum of a certain number of terms in the sequence. For a geometric sequence, where each term follows a consistent mathematical pattern, calculating the partial sum is straightforward. You use a specific formula tailored for this sequence type. The formula for the partial sum of the first \( n \) terms in a geometric sequence is:

• \( S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \)

Here, \( S_n \) is the sum of the first \( n \) terms. \( a_1 \) is the first term. \( r \) is the common ratio.
The partial sum formula helps simplify the calculation, giving you the total of all terms up to a specified number. It's valuable when needing to find the sum quickly without manually adding each term. This is efficient, especially in sequences with larger numbers of terms.

Common Ratio

In any geometric sequence, the common ratio is crucial. It's the factor multiplied by each term to get the next term. Knowing the common ratio lets you understand how the sequence progresses. To determine it, divide a term by the one before it:

• \( r = \frac{a_n}{a_{n-1}} \)

In our exercise, by solving the equation \( r^3 = 8 \), we determined that \( r = 2 \). This means that each term is twice the one before it.
The constant nature of the common ratio uniquely defines geometric sequences. Unlike arithmetic sequences with a constant difference, geometric sequences grow multiplicatively. This property is why geometric sequences rapidly increase or decrease, depending significantly on the value of \( r \).

Geometric Sequence Formula

The geometric sequence formula outlines how to form each term in the sequence. Every term is a result of multiplying the first term by the common ratio raised to a certain power. This formula is:

• \( a_n = a_1 \cdot r^{n-1} \)

Where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( r \) is the common ratio.
Understanding this formula is key to solving any problems involving geometric sequences. It allows you to:

• Find specific terms in the sequence
• Establish relationships between terms
• Derive related formulas like the partial sum formula

During the problem-solving process, you often use this formula to set up equations, as seen with \( a_3 = a_1 \cdot r^2 = 28 \) and \( a_6 = a_1 \cdot r^5 = 224 \). By substituting known values, you can easily solve for unknowns such as \( a_1 \) and \( r \), hence further understanding the sequence.