Simplified Method for Calculating First Term and Common Ratio in Infinite Geometric Series

Hello, math enthusiasts! Today, we’re delving into the captivating realm of infinite geometric series and discovering how to easily determine the “First Term” and the all-important “Common Ratio.” Don’t fret; we’ll simplify the process step by step. By the end of this article, you’ll have a solid grasp of this vital concept.

Understanding Infinite Geometric Series

Before we dive into the simplified method, let’s ensure we’re on the same page regarding infinite geometric series. These are sequences of numbers where each term results from multiplying the preceding term by a constant value. Here’s the general format:

\( S_{\infty} = a + ar + ar² + ar³ + \cdots \)

  • \( S \): The sum of the infinite series.
  • \( a \): The initial term, often referred to as the first term.
  • \( r \): The common ratio.

Comprehending these three components is essential for unravelling the intricacies of infinite geometric series.

Why Finding the First Term & Common Ratio Matters

Now, you might wonder why it’s crucial to uncover the first term and the common ratio. These values serve as the linchpin when tackling infinite geometric series, and here’s why they’re indispensable:

1. Calculating the Sum

Determining the sum of an infinite geometric series hinges on knowing the first term and the common ratio. Without these values, you’d be adrift in a sea of numbers. Grasping these essentials is the initial stride toward efficiently solving these series.

2. Real-World Applications

Infinite geometric series aren’t just theoretical constructs; they manifest in real-life scenarios. They play a role in financial calculations, such as interest rates; they describe phenomena in physics, such as exponential growth or decay, and they’re used in algorithms in computer science. Proficiency in finding the first term and common ratio translates into a valuable skill.

Now that we’ve highlighted their importance, let’s delve into the simplified method for finding these crucial values.

Simplified Method: The Formula-Based Approach

This is the most common and straightforward method for deducing the first term and common ratio. We’ll employ the formula for the sum of an infinite geometric series:

\( \displaystyle S_{\infty} = \frac{a}{1-r} \)

Step 1: Calculate the Sum \(S_{\infty}\)

Initiate the process by determining the sum \(S_{\infty}\) of the infinite series. You may either have this value provided or need to compute it using available information.

Step 2: Rearrange the Formula

Next, rearrange the formula to solve for \(a\), the first term: \(a = S_{\infty} \times (1-r) \)

Step 3: Solve for \(a\)

Plug in the values of \( S_{\infty} \) and \( r \), and calculate \( a \). Voilà, you’ve found the first term!

Let’s illustrate this with an example:

Example: Suppose you have an infinite geometric series with a sum \(S_{\infty} \) of \(20\) and a common ratio \(r\) of \(0.5\). What’s the first term \(a\)?

Solution:

\( \begin{align} a &= 20 \times (1-0.5) \\ &= 20 \times 0.5 \\ &= 10 \end{align} \)

In this series, the first term \(a\) is \(10\).

Simplified Method: The Recursive Approach

While the formula-based approach is effective, there’s another method for determining the first term and common ratio, particularly useful when you already have the first term and need to find the common ratio.

Step 1: Locate the First Term \(a\)

If you possess the first term \(a\), great! You’re already halfway there. If not, you can use the formula-based method explained earlier to calculate \(a\) initially.

Step 2: Find the Common Ratio \(r\)

Now, it’s time to ascertain the common ratio. Here’s the formula: \( \displaystyle r = \frac{\text{second term}}{\text{first term}} \)

Step 3: Verify

Double-check your results by ensuring that each term is indeed the previous term multiplied by the common ratio.

Practical Tips and Tricks

To navigate infinite geometric series with ease, consider these practical tips and tricks:

Tip 1: Embrace Simplicity

When dealing with infinite geometric series, simplification is the key. Strive to express complex fractions or decimals as neat fractions to make your calculations more manageable.

Tip 2: Beware of Negative Ratios

Negative common ratios may introduce alternating signs in your series. Stay vigilant, as this can impact your final sum.

Tip 3: Precision Is Paramount

Mathematics thrives on precision. Follow each step diligently and avoid skipping any. A minor misstep early on can lead to substantial errors later in the process.

Real-Life Examples

Let’s apply our newfound knowledge to real-life scenarios:

1. Financial Investment

Imagine you have a savings account with an initial deposit (first term) of \( \$1,000\), and the bank adds \(5\%\) interest each month (common ratio). How much money will you have in your account after a year?

2. Bacterial Growth

In a laboratory experiment, bacteria double in number every hour. If you start with just one bacterium (first term), how many bacteria will you have after \(24\) hours?

Practice Exercises

Now, it’s your turn to practice. Here are two exercises to sharpen your skills:

Exercise 1:

You have an infinite geometric series with a sum \(S_{\infty} \) of \(80\) and a first term \(a\) of \(20\). Calculate the common ratio \(r\).

Exercise 2:

In an infinite geometric series, the sum \(S_{\infty}\) is \(16\), and the common ratio \(r\) is \(0.25\). Find the first term \(a\).

Feel free to work through these exercises at your own pace. Solutions are provided below:

Exercise 1 Solution:

\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ 1-r &= \frac{a}{S_{\infty}} \\ -r &= \frac{a}{S_{\infty}}-1 \\ r &= 1-\frac{a}{S_{\infty}} \\ &=\frac{S_{\infty}-a}{S_{\infty}} \\ &= \frac{80-20}{80} \\ &= \frac{60}{80} \\ &= 0.75 \end{align} \)

Exercise 2 Solution:

\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ a &= S \times (1-r) \\ &= 16 \times (1-0.25) \\ &= 16 \times 0.75 \\ &= 12 \end{align} \)

In Conclusion

Congratulations! You’ve embarked on a journey into the realm of infinite geometric series and mastered the art of finding the first term and common ratio. These skills transcend mathematics and have practical applications across various fields. Keep practising, and soon you’ll be a maestro of infinite series simplification.

$$ \large \displaystyle S_{\infty} = \frac{\text{first term}}{1-\text{common ratio}} $$

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Question 1

Find the first three terms of the geometric series for which the common ratio \( r=0.6 \) and \( S_{\infty} = 25 \).

\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ \frac{a}{1-0.6} &= 25 \\ \frac{a}{0.4} &= 25 \\ a &= 25 \times 0.4 \\ a &= 10 \\ T_1 &= 10 \\ T_2 &= 10 \times 0.6 \\ T_3 &= 10 \times 0.6^2 \\ \therefore 10, 6 &\text{ and } 3.6 \end{align} \)

Question 2

Find the first three terms of the geometric series for which the first term \( a=10 \) and \( S_{\infty} = 120 \).

\( \displaystyle \begin{align} S_{\infty} &= \frac{a}{1-r} \\ \frac{48}{1-r} &= 120 \\ \frac{1-r}{48} &= \frac{1}{120} \\ 1-r &= \frac{48}{120} \\ -r &= 0.4-1 \\ -r &= -0.6 \\ r &= 0.6 \\ T_1 &=48 \\ T_2 &= 48 \times 0.6 \\ T_3 &= 48 \times 0.6^2 \\ \therefore 48, 28.8 &\text{ and } 17.28 \end{align} \)

Sum to infinity or Limiting Sum of Geometric Series

 

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