Problem 1 - Algebra 2

Question

Find \(S_{n}\) for each geometric series described. $$ a_{1}=5, r=2, n=14 $$

Step-by-Step Solution

Answer

The sum of the first 14 terms, \(S_{14}\), is 81915.

Step 1: Identify the Formula

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

Step 2: Plug in the Values of the Series

Substitute the given values into the formula: \( a_1 = 5 \), \( r = 2 \), and \( n = 14 \). Thus, the equation becomes:\[S_{14} = 5 \frac{1-2^{14}}{1-2}\]

Step 3: Simplify the Denominator

Calculate the denominator: \(1 - r = 1 - 2 = -1\).

Step 4: Calculate the Numerator

Find \(1 - 2^{14}\): First, calculate \(2^{14} = 16384\). Then, compute \(1 - 16384 = -16383\).

Step 5: Divide the Numerator by the Denominator

Divide the calculated numerator by the denominator: \[\text{Numerator: } -16383 \quad \text{Denominator: } -1\]Thus, \(-16383 / -1 = 16383\).

Step 6: Multiply by the First Term

Finally, multiply the result from Step 5 by \(a_1\):\[S_{14} = 5 \times 16383 = 81915\]

Step 7: Conclusion

The sum of the first 14 terms of the geometric series is 81915.

Key Concepts

Sum of Geometric Series

The concept of the sum of a geometric series is essential in understanding sequences and mathematics. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. To find the sum of the first \( n \) terms of a geometric series, we use a special formula:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]Here, \( S_n \) represents the sum of the series, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula helps us calculate the total value of all terms up to the \( n \)-th in the sequence. It's particularly useful in various fields like finance and engineering, where such calculations are frequent.

• Allows for easy calculation of large series
• Important for solving problems involving repeated multiplication

Common Ratio

The common ratio in a geometric series is the constant factor by which each term is multiplied to get the next term. In mathematical notation, it is denoted by \( r \). Calculating the common ratio is simple: divide any term in the series by the term that comes before it.For example, in a series where the first few terms are 5, 10, 20, and 40, the common ratio \( r \) would be found as:

• \( 10/5 = 2 \)
• \( 20/10 = 2 \)
• \( 40/20 = 2 \)

In the given exercise, the common ratio is 2. This indicates that each term is twice the previous one. Understandably, the common ratio plays a pivotal role in determining the growth or decrease pattern in the series.

First Term

The first term of a geometric series, denoted by \( a_1 \), acts as the starting point for the entire series. It is the initial value before the common ratio is applied successively to find subsequent terms.This initial value is crucial because it's the baseline from which the series grows or diminishes, depending on whether the common ratio is greater than or less than one. The formula for the sum of a geometric series specifically uses this first term to compute the total sum:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]To give an example with our exercise, the first term \( a_1 = 5 \) sets the series in motion. All subsequent operations in the given sequence rely on this value being used as the initial starting point. This is why the first term can drastically affect the sum of a series.

Number of Terms

The number of terms, represented by \( n \), in a geometric series determines how many terms are summed up using the sum formula. It tells us how deep into the series we go, adding each successive term as dictated by the common ratio.In our exercise, wherein \( n = 14 \), it means we're considering the sum of the first 14 terms in the sequence.

• Directly affects the exponent of \( r \) in the sum formula
• Each term's value increases or decreases based on the common ratio and this count

Using a larger \( n \) can result in significant increases in the sum if the common ratio is greater than one. Alternatively, a smaller \( n \) or a common ratio less than one would yield a smaller total when summing the series.