Problem 24 - Algebra for College Students

Question

Find the sum of the first ten terms of the sequence where \(a_{n}=2^{5-n} .31 \frac{31}{32}\)

Step-by-Step Solution

Answer

The sum of the first ten terms is 31.

Step 1: Identify the sequence type

The given sequence is defined by the formula \(a_{n}=2^{5-n}\). It represents a geometric sequence because each term is obtained by multiplying the previous term by a constant factor.

Step 2: Recognize geometric sequence properties

For a geometric sequence, a common ratio \(r\) can be found by dividing any term by the previous term. Here, the common ratio \(r=\frac{a_{n+1}}{a_n}=\frac{2^{5-(n+1)}}{2^{5-n}}=\frac{1}{2}\).

Step 3: Identify the first term

To find the sum of the first ten terms, begin by identifying the first term of the sequence when \(n=1\). Substitute \(n=1\) into the formula: \(a_1=2^{5-1}\cdot\frac{31}{32}=16\cdot\frac{31}{32}=15.5\).

Step 4: Use the formula for the sum of a geometric series

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by: \[S_n = a_1 \cdot \frac{1-r^n}{1-r}\] where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 5: Substitute values to find the sum

Now, substitute \(a_1=15.5\), \(r=\frac{1}{2}\), and \(n=10\) into the formula: \[S_{10} = 15.5 \cdot \frac{1-(\frac{1}{2})^{10}}{1-\frac{1}{2}}\] Calculate accordingly: \[S_{10} = 15.5 \cdot \frac{1-\frac{1}{1024}}{1-\frac{1}{2}} = 15.5 \cdot \frac{1023/1024}{1/2} = 15.5 \cdot \frac{1023}{512}\].

Step 6: Calculate the sum

Perform the calculation from the previous step: \[S_{10} = 15.5 \times \frac{1023}{512} = 31\]. Thus, the sum of the first ten terms of the sequence is 31.

Key Concepts

Sum of Geometric Series

Knowing how to find the sum of a geometric series is a crucial skill in mathematics, especially when dealing with sequences. For a geometric sequence, each term has a consistent ratio, known as the common ratio, multiplied to the previous term. To calculate the sum of the first few terms in a geometric series, we use the formula: \[ S_n = a_1 \cdot \frac{1-r^n}{1-r} \] where \(a_1\) represents the first term, \(r\) is the common ratio, and \(n\) is the total number of terms. This formula helps us quickly find the sum without having to add each term individually. Let's break it down further:

• First Term (\(a_1\)): The initial term of the sequence.
• Common Ratio (\(r\)): The consistent factor by which each term is multiplied to get to the next.
• Terms (\(n\)): The number of terms you want to include in the sum.

Using this knowledge, we can efficiently determine the sum of terms in a sequence with ease.

Common Ratio in Sequences

The common ratio in a geometric sequence is the factor by which each term is multiplied to obtain the next term. It's the essence of geometric progression. Finding this ratio is straightforward: choose any term in the sequence and divide it by the preceding term. This gives us the ratio that forms the backbone of the sequence. In the given exercise, for example, the common ratio \(r\) was determined as \(\frac{1}{2}\). Here’s how you can verify:

• Take the term formula: \(a_n = 2^{5-n} \cdot \frac{31}{32}\), where \(n = 2\) gives us the second term.
• Divide the second term by the first to obtain: \(-)\) \(\frac{1}{2}\).

Thus, once you have the common ratio, predicting future or past terms becomes a simple multiplication or division task. Understanding this concept makes working with geometric sequences much more intuitive.

First Term of a Sequence

In any sequence, the first term plays a significant role as it sets the starting value and influences subsequent terms. In arithmetic and geometric sequences, this term often determines the overall trend and direction of the sequence. To find the first term \(a_1\) of a sequence, you substitute the starting value of the index, typically \(n = 1\), into the sequence formula. From our exercise:

• We substituted \(n = 1\) into \(a_n = 2^{5-n} \cdot \frac{31}{32}\).
• This gave us \(a_1 = 15.5\), setting the initial anchor for our sequence.

The first term is essential when calculating the sum of terms, as seen in our formula for the sum of a geometric series. Without the correct first term, all subsequent calculations could be skewed. Therefore, mastering the ability to identify and work with the first term is foundational in sequence-related problems.