Problem 39 - Precalculus: Mathematical for Calculus

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 is 315.

Step 1: Identify the Given Variables

We're given the first term of the geometric sequence, denoted as \(a = 5\), the common ratio \(r = 2\), and the number of terms \(n = 6\). Our goal is to find the partial sum \(S_n\) of this sequence.

Step 2: Recall the Formula for the Partial Sum

The formula for the partial sum \(S_n\) of the first \(n\) terms of a geometric sequence is \(S_n = a \frac{(r^n - 1)}{r - 1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 3: Substitute Known Values into the Formula

Substitute the given values into the partial sum formula: \(S_6 = 5 \frac{(2^6 - 1)}{2 - 1}\). This simplifies the expression for calculation.

Step 4: Calculate the Power of the Common Ratio

Calculate \(2^6\), which equals \(64\). Therefore, the expression inside the parentheses becomes \(2^6 - 1 = 64 - 1 = 63\).

Step 5: Evaluate the Denominator

Calculate the denominator of the fraction, which is \(r - 1 = 2 - 1 = 1\).

Step 6: Complete the Calculation

Substitute these evaluations back into the expression: \(S_6 = 5 \frac{63}{1} = 5 \times 63 = 315\).

Step 7: Final Answer

The partial sum \(S_6\) of the geometric sequence with the given conditions is \(315\).

Key Concepts

Partial Sum

In the study of sequences, the partial sum of a geometric sequence is an important concept. It refers to the sum of a certain number of terms from the sequence. Specifically, if you're given a geometric sequence, the partial sum, denoted as \(S_n\), is the sum of the first \(n\) terms.When calculating a partial sum, you're adding together the values of the sequence from the first term up through the \(n^{th}\) term. It's like finding out how much you've accumulated if you keep adding numbers that follow a particular multiplying pattern (common ratio) starting from the first term.In mathematical notation, the partial sum for a geometric sequence is represented by the formula:

• \(S_n = a \frac{(r^n - 1)}{r - 1}\)

where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms you want to sum.

Common Ratio

The common ratio in a geometric sequence is the constant factor by which each term is multiplied to get the next term. This is what distinguishes a geometric sequence from other types of sequences, such as arithmetic sequences where the difference between terms is constant.It is represented by the symbol \(r\). To calculate the common ratio, you take any term in the sequence and divide it by the term before it. For example, if the first few terms of a sequence are 2, 6, 18, ..., the common ratio \(r\) is:

• \(r = \frac{6}{2} = 3\)

A quick point to note is that if the common ratio is greater than 1, the terms will increase exponentially. If it's between 0 and 1, the terms will decrease, and if it's negative, the sequence will alternate in sign.

First Term

The first term of a geometric sequence is crucial because it serves as the starting point for the sequence. It is commonly represented by the letter \(a\).In a geometric sequence, knowing the first term \(a\) is necessary to determine all subsequent terms when the common ratio \(r\) is given. The sequence is defined by multiplying the first term \(a\) by the common ratio \(r\) to find the second term, continuing this pattern for succeeding terms.For example, if the first term \(a = 5\) and the common ratio \(r = 2\), the first three terms of the sequence would be:

• 1st term: \(5\)
• 2nd term: \(5 \times 2 = 10\)
• 3rd term: \(10 \times 2 = 20\)

Understanding the first term helps in identifying or reconstructing the entire sequence.

Geometric Sequence Formula

The formula for a geometric sequence is the backbone for understanding how these sequences operate. This formula allows you to determine any term in the sequence if the first term and the common ratio are known.The general formula for the \(n^{th}\) term of a geometric sequence is given by:

• \(a_n = a \cdot r^{n-1}\)

where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number you are interested in. Using this formula, you can calculate any term within the sequence efficiently.If you want to know more than just an individual term, such as a sum over several terms (i.e., a partial sum), you would use the partial sum formula:

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

This formula shows how interconnected the sequence elements are through the common ratio.