Problem 41 - Contemporary Precalculus

Question

In Exercises \(39-42,\) find the kth partial sum of the geometric sequence \(\left\\{a_{n}\right\\}\) with common ratio \(r\). $$k=7, a_{2}=6, r=2$$

Step-by-Step Solution

Answer

Answer: The 7th partial sum of the geometric sequence is 381.

Step 1: Find the first term of the sequence (\(a_1\))

Since we have the second term of the sequence, \(a_2 = 6\), and the common ratio, \(r = 2\), we can use the formula to find the first term, \(a_1\): $$a_2 = a_1 \cdot r$$ Plugging the given values, we get: $$6 = a_1 \cdot 2$$ Now we can solve for \(a_1\): $$a_1 = \frac{6}{2} = 3$$ So the first term, \(a_1 = 3\).

Step 2: Calculate the 7th partial sum of the sequence (\(S_7\))

Now that we know the first term, \(a_1 = 3\), and the common ratio, \(r = 2\), we can use the formula for the kth partial sum to find \(S_7\): $$S_k = \frac{a_1\left(1 - r^k\right)}{1 - r}$$ Substituting the given values: $$S_7 = \frac{3\left(1 - 2^7\right)}{1 - 2}$$ Calculate the terms within the parentheses: $$S_7 = \frac{3\left(1 - 128\right)}{-1}$$ Now, simplify the expression: $$S_7 = \frac{3\left(-127\right)}{-1}$$ Finally, perform the calculations: $$S_7 = \frac{-381}{-1} = 381$$ The 7th partial sum of the sequence is \(S_7 = 381\).

Key Concepts

Understanding Partial Sum

A partial sum in a geometric sequence refers to the sum of a specific number of terms, starting from the first term up to a particular term. It is a crucial concept for calculating and understanding the behavior of a sequence.
For example, if we want to find the 7th partial sum of a sequence, we sum up the first seven terms. This is particularly useful because it allows us to analyze a subset of the entire sequence.
The formula for finding the partial sum of a geometric sequence is given by:

• \( S_k = \frac{a_1(1 - r^k)}{1 - r} \)

where \( S_k \) is the partial sum, \( a_1 \) is the first term, \( r \) is the common ratio, and \( k \) is the number of terms to be summed.
The formula helps simplify the calculation, especially when dealing with large sequences, saving time and effort.

Decoding Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. This consistency makes geometric sequences predictable and easy to analyze.
To find the common ratio, divide any term by the previous term:

• \( r = \frac{a_{n+1}}{a_n} \)

In our exercise, the common ratio is \( r = 2 \), meaning each term is double the previous term.
This understanding is vital because once the common ratio is known, you can easily find any term in the sequence using the formula:

• \( a_n = a_1 \cdot r^{n-1} \)

Knowing the common ratio also allows for the calculation of the partial sum, as it is used in its formula.

Exploring Sequence Terms

Sequence terms in a geometric sequence are the individual elements of the sequence, each derived from multiplying the previous term by the common ratio.
In a sequence defined by \( a_1 = 3 \) and \( r = 2 \), the first few terms are:

• \( a_1 = 3 \)
• \( a_2 = a_1 \cdot r = 3 \cdot 2 = 6 \)
• Continue this process: \( a_3 = a_2 \cdot r = 12 \)

The pattern continues, and each term is derived from the preceding one.
Understanding sequence terms is key to comprehending the entire structure of a geometric sequence.
This sequential growth by a common ratio is what distinguishes geometric sequences from other types of sequences, such as arithmetic ones, where the difference is constant instead of the ratio.