Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+a_{n}, n=6 $$

Step-by-Step Solution

Answer

The sum of the first 6 terms is \( \frac{63}{64} \).

Step 1: Identify the First Term and Common Ratio

In a geometric series, each term after the first is obtained by multiplying the previous term by a constant known as the common ratio. Here, the first term \( a_1 \) is \( \frac{1}{2} \) and the common ratio \( r \) can be found by dividing the second term by the first term: \( r = \frac{1}{4} \div \frac{1}{2} = \frac{1}{2} \).

Step 2: Use the Formula for the Sum of the First \(n\) Terms of a Geometric Series

The sum \(S_n\) of the first \(n\) terms of a geometric series can be calculated using the formula: \[ S_n = a_1 \frac{1 - r^n}{1-r} \] Substitute \(a_1 = \frac{1}{2}\), \(r = \frac{1}{2}\), and \(n = 6\) into the formula.

Step 3: Calculate \( r^n \)

Calculate \( (\frac{1}{2})^6 \): \[ (\frac{1}{2})^6 = \frac{1}{2^6} = \frac{1}{64} \].

Step 4: Substitute Values into the Sum Formula

Substitute \( a_1 = \frac{1}{2} \), \( r^6 = \frac{1}{64} \), and \( r = \frac{1}{2} \) into the formula to get: \[ S_6 = \frac{1}{2} \cdot \frac{1 - \frac{1}{64}}{1 - \frac{1}{2}} \] This simplifies to: \[ S_6 = \frac{1}{2} \cdot \frac{63}{64} \cdot 2 \].

Step 5: Simplify the Expression

Simplify the expression obtained in the previous step: \[ S_6 = \frac{63}{64} \].

Step 6: Interpret the Result

The sum of the first 6 terms of the geometric series is \( \frac{63}{64} \).

Key Concepts

Sum of n terms

When dealing with a geometric series, one of the key tasks is to find the sum of its terms. If you've started learning about these series, don't worry; it's simpler than it seems!
To find the sum of the first \(n\) terms of a geometric series, we use the formula:

• \( S_n = a_1 \frac{1 - r^n}{1-r} \)

Here, \(S_n\) represents the sum of the first \(n\) terms.
The term \(a_1\) stands for the first term in the series, and \(r\) stands for the common ratio.
The formula calculates the total by adjusting for the common ratio. This only works if the absolute value of \(r\) is less than 1, which is common in these problems.
By plugging the appropriate values into this formula, such as \(a_1 = \frac{1}{2}\), \(r = \frac{1}{2}\), and \(n = 6\), you can compute the exact sum of the given series.

Common Ratio

The common ratio is a fundamental part of understanding geometric sequences and series.
It is the factor by which we multiply each term to get to the next term in the sequence.

• For instance, if your first term is \(\frac{1}{2}\) and your second term is \(\frac{1}{4}\), the common ratio \(r\) can be calculated by dividing the second term by the first term, which is \(\frac{1}{4} \div \frac{1}{2} = \frac{1}{2}\).

Understanding the value of the common ratio helps in determining the behavior of the series.
If \(|r| < 1\), the terms in the series get smaller and the series converges; if \(|r| > 1\), the terms grow larger. In our example, \(r = \frac{1}{2}\) allows us to use the formula to find the sum of the series.

Geometric Sequence Formula

Every geometric sequence can be described using its specific formula. This takes any sequence and provides a way to determine any term within it without writing out all previous terms.
The formula we use is:

• \( a_n = a_1 \times r^{(n-1)} \)

In this formula, \(a_n\) is the \(n\)-th term you're trying to find.
The \(a_1\) is the first term, and \(r\) is the common ratio.
For instance, if you want to find the 6th term in a sequence where each magic number repeats its magic factor, with \(a_1\) being \(\frac{1}{2}\) and \(r = \frac{1}{2}\), you plug these into the equation to find \(a_6\).
Using such a formula is especially useful for identifying terms without step-by-step counting or when finding terms far along in the series.