3. Sum of the First 𝑛 Terms (Finite Geometric Series)

The sum of the first 𝑛 terms of a geometric sequence is given by:

📌 Formula (when 𝑟 ≠ 1):
𝑆 = $a \cdot \frac{1 - r^n}{1 - r}$

In this chunk, we learn how to calculate the sum of the first𝑛 terms of a geometric sequence. The formula provided is crucial as it gives a way to quickly find the total of a specific number of terms without having to add each term individually. Here, $S$ represents the total sum, $a$ is the first term of the sequence, $r$ is the common ratio, and $n$ is the number of terms being summed. The formula only works when the common ratio $r$ is not equal to 1 because if it were, every term would be the same, rendering the formula meaningless as it would lead to division by zero.

Think of a savings account where you make regular deposits. If you continue to add the same amount each period, the total amount grows rapidly due to the interest compounding on each deposit. This is what the finite geometric series formula represents, allowing you to calculate how much you've saved after a set number of periods based on your initial deposit and the interest rate.

✅ Example 2:
Find the sum of the first 5 terms of the sequence:
𝑎 = 2, 𝑟 = 3
Solution:
$S = 2 \cdot \frac{1 - 3^5}{1 - 3}$
$S = 2 \cdot \frac{1 - 243}{-2}$
$S = 2 \cdot \frac{-242}{-2}$
$S = 2 \cdot 121 = 242$

In this example, we apply the formula for the sum of the first 5 terms with the first term as 2 and the common ratio as 3. First, we use the formula, substituting $a$ and $r$. Next, we calculate $3^5$, which means multiplying 3 by itself a total of 5 times, resulting in 243. Hence, we find $1 - 3^5 = 1 - 243 = -242$. The denominator becomes $1 - 3 = -2$. Thus, we can simplify the sum to yield $S = 2 \cdot \frac{-242}{-2} = 2 \cdot 121 = 242$. Therefore, the sum of the first 5 terms is 242.

Imagine a bank account where you invest a fixed amount of money that grows every year. Each year's interest earnings (your common ratio) increase the total amount at a faster rate. Calculating how much you’d have after a certain number of deposits gives you a clear picture, just like we see in this example where we found the sum of the first 5 terms.

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