Geometric Series Sum: First 8 Terms

Let's dive into the fascinating world of geometric sequences and series! Today, we're going to tackle a problem that involves finding the sum of the first 8 terms of a specific sequence: $100, 121, 146.41, ext{ extellipsis}$. This is a classic example of a geometric series, and understanding how to calculate its sum is a fundamental skill in mathematics. We'll be using the well-known formula for the sum of a finite geometric series, which is S_n=rac{a_1 ext{}(1-r^n ext{})}{1-r}. Don't worry if the formula looks a bit daunting at first; we'll break it down step by step. Our goal is to find the sum of the initial eight numbers in this sequence, and we'll make sure to round our final answer to the nearest hundredth if any decimals pop up. So, grab your calculators, and let's embark on this mathematical journey together! We'll start by identifying the key components of the sequence, namely the first term ($a_1$) and the common ratio ($r$), which are crucial for plugging into our formula. The sequence itself provides these clues directly, and once we have them, the rest is just a matter of careful calculation. We'll also touch upon why this formula works, giving you a deeper appreciation for the mathematics behind it. Understanding geometric series is not just about solving problems; it's about grasping patterns and growth, which have applications in finance, science, and many other fields.

Unpacking the Geometric Sequence: Finding $a_1$ and $r$

Before we can use the formula S_n=rac{a_1 ext{}(1-r^n ext{})}{1-r} to find the sum of the first 8 terms, we first need to identify the critical values from our given sequence: $100, 121, 146.41, ext{ extellipsis}$. The first term, denoted as $a_1$, is straightforward. It's simply the very first number in the sequence. In this case, $a_1 = 100$. Now, let's find the common ratio, $r$. The common ratio is the constant factor by which each term is multiplied to get the next term. To find it, we can divide any term by its preceding term. Let's try dividing the second term by the first term: r = rac{121}{100} = 1.21. To confirm that this is indeed a geometric sequence with a common ratio of 1.21, let's check if multiplying the second term by 1.21 gives us the third term: $121 imes 1.21 = 146.41$. It does! This confirms that our common ratio is $r = 1.21$. We also know that we need to find the sum of the first 8 terms, so our $n$ value is $n = 8$. With these three key values – $a_1 = 100$, $r = 1.21$, and $n = 8$ – we are now fully equipped to plug them into the sum formula and calculate the desired sum. This initial step of identifying the sequence's parameters is fundamental to solving any geometric series problem. It's like gathering the ingredients before you start baking – you need to know what you're working with to achieve the desired outcome. The process is simple division, but the importance of verifying the ratio by checking subsequent terms cannot be overstated, ensuring we are dealing with a true geometric progression.

Applying the Geometric Series Formula for the Sum

Now that we've identified our key values – $a_1 = 100$, $r = 1.21$, and $n = 8$ – we can confidently apply the formula for the sum of a finite geometric series: S_n=rac{a_1 ext{}(1-r^n ext{})}{1-r}. Our goal is to find $S_8$, the sum of the first 8 terms. Let's substitute our values into the formula:

S_8=rac{100 ext{}(1-(1.21)^8 ext{})}{1-1.21}

The first step is to calculate $(1.21)^8$. Using a calculator, we find that $(1.21)^8 ext{ extapprox } 5.166971821881$. Now, let's substitute this back into the formula:

S_8=rac{100 ext{}(1-5.166971821881 ext{})}{1-1.21}

Next, we calculate the numerator: $1 - 5.166971821881 = -4.166971821881$. So, the numerator becomes $100 imes (-4.166971821881) = -416.6971821881$.

Now, let's calculate the denominator: $1 - 1.21 = -0.21$.

Finally, we divide the numerator by the denominator:

S_8=rac{-416.6971821881}{-0.21}

$S_8 ext{ extapprox } 1984.2722961338$

We are asked to round the result to the nearest hundredth. Looking at the third decimal place (2), we see that it is less than 5, so we round down. Therefore, the sum of the first 8 terms of the sequence is approximately $1984.27$. This calculation demonstrates the power of the geometric series formula in efficiently summing up a sequence, especially when dealing with a larger number of terms. It avoids the tedious process of calculating each term individually and then adding them up.

Understanding the Formula: Why Does It Work?

It's always good to understand why a formula works, rather than just blindly applying it. The formula for the sum of a finite geometric series, S_n=rac{a_1 ext{}(1-r^n ext{})}{1-r}, is derived through a clever algebraic manipulation. Let's write out the sum of the first $n$ terms of a geometric sequence:

$S_n = a_1 + a_1r + a_1r^2 + ext{ extellipsis} + a_1r^{n-1}$

Now, multiply both sides of this equation by the common ratio, $r$:

$rS_n = a_1r + a_1r^2 + a_1r^3 + ext{ extellipsis} + a_1r^n$

Observe that most of the terms in $S_n$ and $rS_n$ are the same. Let's subtract the second equation from the first one:

$S_n - rS_n = (a_1 + a_1r + a_1r^2 + ext{ extellipsis} + a_1r^{n-1}) - (a_1r + a_1r^2 + a_1r^3 + ext{ extellipsis} + a_1r^n)$

Many terms cancel out! We are left with:

$S_n - rS_n = a_1 - a_1r^n$

Now, we can factor out $S_n$ on the left side and $a_1$ on the right side:

$S_n(1 - r) = a_1(1 - r^n)$

Finally, to isolate $S_n$, we divide both sides by $(1 - r)$, assuming $r eq 1$ (if $r=1$, the sum is simply $n imes a_1$ because all terms are the same):

S_n = rac{a_1(1 - r^n)}{1 - r}

This derivation shows that the formula is a direct consequence of the definition of a geometric sequence and basic algebra. It elegantly pairs terms to eliminate intermediate values, leaving only the first term and the last term (in a modified form) to determine the total sum. This mathematical elegance is a hallmark of many useful formulas in mathematics, making complex calculations surprisingly manageable.

Conclusion: The Power of Geometric Series

In this exploration, we successfully calculated the sum of the first 8 terms of the geometric sequence $100, 121, 146.41, ext{ extellipsis}$, arriving at a result of approximately $1984.27$. We achieved this by first identifying the first term ($a_1 = 100$) and the common ratio ($r = 1.21$), and then applying the formula for the sum of a finite geometric series, S_n=rac{a_1 ext{}(1-r^n ext{})}{1-r}. We also took a moment to understand the algebraic derivation of this powerful formula, reinforcing our grasp of its underlying principles. Geometric series are more than just abstract mathematical concepts; they represent patterns of growth and decay that are fundamental in many real-world applications. From compound interest in finance, where investments grow by a fixed percentage each period, to the study of population dynamics or radioactive decay in science, the principles of geometric sequences and series are ubiquitous. Understanding how to calculate their sums allows us to predict future values, analyze trends, and make informed decisions. The ability to efficiently sum these series means we can model and understand phenomena that exhibit exponential change without needing to calculate each individual step. This mathematical tool is invaluable for anyone looking to understand growth and accumulation over time.

For further exploration into the fascinating world of sequences and series, you can visit Khan Academy's Mathematics section, a fantastic resource for learning and mastering mathematical concepts.