Problem 45 - Algebra and Trigonometry

Question

Find the sum. $$ \sum_{k=0}^{10} 3\left(\frac{1}{2}\right)^{k} $$

Step-by-Step Solution

Answer

The sum is approximately 6.

Step 1: Identify the Type of Series

The given expression represents a geometric series with the general term formula \( a_k = 3 \left(\frac{1}{2}\right)^k \). The series is finite, ranging from \( k = 0 \) to \( k = 10 \).

Step 2: Identify the Common Ratio and First Term

The first term of the series, \( a_0 \), is found by substituting \( k = 0 \) into the general term, which gives \( a_0 = 3 \times 1 = 3 \). The common ratio \( r \) is \( \frac{1}{2} \).

Step 3: Use the Formula for the Sum of a Finite Geometric Series

For a geometric series of the form \( \sum_{k=0}^{n} a r^k \), the sum is calculated using the formula:\[ S_n = a \frac{1-r^{n+1}}{1-r} \]Where:- \( a = 3 \) (first term)- \( r = \frac{1}{2} \) (common ratio)- \( n = 10 \) (number of terms minus one)

Step 4: Substitute Values into the Sum Formula

Substitute \( a = 3 \), \( r = \frac{1}{2} \), and \( n = 10 \) into the formula:\[ S_{10} = 3 \frac{1-(\frac{1}{2})^{11}}{1-\frac{1}{2}} \]

Step 5: Simplify the Expression

Calculate \( r^{n+1} = \left(\frac{1}{2}\right)^{11} = \frac{1}{2048} \).Now, simplify the formula step by step:\[ S_{10} = 3 \frac{1 - \frac{1}{2048}}{1-\frac{1}{2}} = 3 \frac{\frac{2047}{2048}}{\frac{1}{2}} \]

Step 6: Further Simplification

Multiply the terms:\[ S_{10} = 3 \times 2 \times \frac{2047}{2048} = 6 \times \frac{2047}{2048} \]

Step 7: Calculate the Numerical Result

Calculate the above expression:\( S_{10} = \frac{12282}{2048} \)Divide both the numerator and the denominator by 2 to simplify:\( S_{10} = \frac{6141}{1024} \approx 5.996 \), which further simplifies to \( S_{10} \approx 6 \).

Step 8: Verify and Conclude

Confirm the calculation is consistent with the geometric series formula.

Key Concepts

Finite Series

A finite series is a sum of a sequence of numbers where the sequence has a defined starting point and a finite endpoint. This means that the number of terms in the series is countable, and you can identify the first and the last terms. In a finite series, each term follows a particular rule or pattern that allows us to calculate the sum more straightforwardly. This is unlike infinite series, which continue indefinitely and require more complex methods to find their sum. In the given problem, we are dealing with a finite geometric series that ranges from the term with index 0 to the term with index 10. This explicit range means there are a total of 11 terms in the series to be summed up.

Common Ratio

The common ratio is a key feature of a geometric sequence or series. It is the factor by which we multiply one term in the sequence to get to the next term. To maintain the properties of a geometric sequence, it's crucial that this ratio remains constant throughout. Identifying the common ratio allows us to predict the behavior of the terms within the sequence.
For the series in our exercise, the common ratio is given as \( \frac{1}{2} \). This means that each term is half the previous term, clearly showing the geometric nature of the progression. By multiplying each term by the common ratio, we seamlessly generate the complete series. Determining the common ratio is vital before calculating any geometric series sum.

Sum Formula

To find the sum of a finite geometric series, we use a special formula designed for this purpose. The sum formula allows us to calculate the entire sum without the need to manually add each term. It is given by:

• \( S_n = a \frac{1-r^{n+1}}{1-r} \)
• where \( S_n \) is the sum of the first \( n+1 \) terms, \( a \) is the first term, and \( r \) is the common ratio.

This formula subtracts the respective power of the common ratio raised to one plus the number of terms from 1 and divides the result by \( 1-r \). In our case, substituting the values \( a = 3 \), \( r = \frac{1}{2} \), and \( n = 10 \) into the formula yielded the series sum. Understanding this formula is essential as it simplifies finding the sum of any finite geometric series.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The property that distinguishes geometric sequences is this fixed multiplicative factor.
In our example, we observe that the sequence begins with the term 3, and each subsequent term is formed by multiplying the previous term by \( \frac{1}{2} \). This consistent pattern of multiplication creates the sequence:

• 3,
• \( 3 \times \frac{1}{2} = 1.5 \),
• \( 1.5 \times \frac{1}{2} = 0.75 \),
• ... and so forth.

Recognizing a sequence as geometric is highly useful since it allows us to employ the sum formula efficiently and accurately in various applications. With geometric sequences, predicting future terms and computing series sums become manageable tasks.