Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=4, a_{5}=324, n=9 $$

Step-by-Step Solution

Answer

The sum of the series is 39364.

Step 1: Identify Given Information

From the problem, we know the first term \(a_1 = 4\), the fifth term \(a_5 = 324\), and the number of terms \(n = 9\).

Step 2: Find the Common Ratio

The formula for the \(n\)-th term in a geometric series is \(a_n = a_1 \, r^{n-1}\). Use this with \(a_5 = 324\) to find the common ratio \(r\):\[324 = 4 \, r^{4} \]Solve for \(r\) by dividing both sides by 4:\[r^4 = \frac{324}{4} = 81\]\(\Rightarrow r = 81^{1/4} = 3\). The common ratio is \(r = 3\).

Step 3: Use the Formula for the Sum of the Series

The formula for the sum of the first \(n\) terms of a geometric series is:\[S_n = \frac{a_1(1 - r^n)}{1 - r}\]Since \(r = 3\), \(a_1 = 4\), and \(n = 9\), substitute these values into the formula:\[S_n = \frac{4(1 - 3^9)}{1 - 3}\]

Step 4: Calculate the Sum

Calculate \(3^9\):\[3^9 = 19683\]Substitute this back into the sum formula:\[S_9 = \frac{4(1 - 19683)}{-2}\]Calculate further:\[S_9 = \frac{4(-19682)}{-2} = \frac{-78728}{-2} = 39364\]

Step 5: Conclusion

The sum of the first 9 terms of the geometric series is 39364.

Key Concepts

Sum of Geometric Series

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. When you need to find the sum of a specific number of terms in such a series, use the handy geometric series sum formula. This can save you a lot of tedious calculation work, especially if you have many terms!

The formula to find the sum of the first \( n \) terms in a geometric series is:

• \( S_n = \frac{a_1(1 - r^n)}{1 - r} \)

This formula is crucial because it allows you to calculate the total of these terms without listing and adding each one individually. Note, if the absolute value of the common ratio \(|r| < 1\), the series converges. Otherwise, it can grow or shrink rapidly, depending on the sign and magnitude of \( r \).

In our problem, having \( a_1 = 4 \), \( r = 3 \), and \( n = 9 \), simply plug these into the formula, which calculates the sum of the series as 39364 after evaluating \(3^9\) as 19683, performing subsequent multiplications, and simplifications!

Common Ratio in Sequences

In every geometric progression, there's a key element called the common ratio. This ratio is the factor that connects each term to the next, and you find it by dividing any term by the term before it.

In our specific exercise, we learned the common ratio \( r \) inside the problem was determined by using the formula for \( n \)-th term in a geometric sequence, \( a_n = a_1 \, r^{n-1} \). From the given values \( a_5 = 324 \) and \( a_1 = 4 \), we calculated:

• \( 324 = 4 \, r^4 \)
• This simplifies to \( r^4 = 81 \), which implies \( r = 81^{1/4} = 3 \).

Knowing how to find the common ratio is fundamental because it describes how the sequence evolves. Each term is simply the previous term multiplied by this common ratio. Once we have this, solving any related problem becomes much more straightforward! More so, this understanding is essential when both constructing and deconstructing geometric sequences for any mathematical or real-world application.

Geometric Progression Formula

A geometric progression is a series where each term increases (or decreases) by multiplying the previous term by a constant. This consistency is the defining characteristic of such sequences. The general form of the formula for the \( n \)-th term is:

• \( a_n = a_1 \, r^{n-1} \)

In this expression:

• \( a_n \) is the \( n \)-th term
• \( a_1 \) the first term
• \( r \) the common ratio

Each new term is found by taking the first term and multiplying by the common ratio raised to the power of \( n-1 \).

Understanding how to use the geometric progression formula helps you determine any term within a sequence quickly. In our problem, we used this formula to confirm that the sequence and find specific terms efficiently. Grasping these fundamentals gives you the knowledge to handle any geometric progression challenge, making solving problems related to both sequences and series that much easier.