Problem 61 - Beginning and Intermediate Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{5}(-4)\left(3^{i}\right)$$

Step-by-Step Solution

Answer

The sum of the first 5 terms of the given geometric sequence is \(S_5 = -484\).

Step 1: Identify the values of a, r, and n

In this case, we are given: - First term, \(a = -4\) - Common ratio, \(r = 3\) - Number of terms to be summed, \(n = 5\)

Step 2: Use the formula for the sum of a geometric sequence

Now, we will use the formula for the sum of the first n terms of a geometric sequence: $$S_n = \sum_{i=1}^{n} ar^{i-1}$$ Plug in the values of \(a\), \(r\), and \(n\) that we identified in Step 1: $$S_5 = \sum_{i=1}^{5}((-4) \cdot 3^{i-1})$$

Step 3: Calculate the sum

Next, we calculate the sum: $$S_5 = -4(3^{0}) + (-4)(3^{1}) + (-4)(3^{2}) + (-4)(3^{3}) + (-4)(3^{4})$$ $$S_5 = -4(1) - 4(3) - 4(9) - 4(27) - 4(81)$$ $$S_5 = -4 - 12 - 36 - 108 - 324$$ $$S_5 = -484$$ Therefore, the sum of the first 5 terms of the given geometric sequence is -484.

Key Concepts

sum of terms

When dealing with a geometric sequence, finding the sum of its terms can sometimes seem challenging. However, there's an effective formula designed specifically for that purpose. The sum of a geometric sequence's first n terms is denoted as \( S_n \), and it represents the total of all the terms from the start up to the nth term. In practical terms, you start with the first term and keep adding each subsequent term, which are obtained by multiplying by the common ratio, all the way until you reach the nth term. By using the geometric sequence formula, which simplifies this process, you can quickly calculate the sum without manually adding each term individually.

common ratio

The **common ratio** is a pivotal element in any geometric sequence. It dictates how each term in the sequence relates to the one before it. To find the common ratio, you divide any term in the sequence by the previous term. In the exercise, our sequence is affected by the common ratio of 3. This means each subsequent term is 3 times the previous term. Understanding and identifying the common ratio allows a grasp of the sequence's growth or decline pattern, which is crucial for calculating the sum of terms in the sequence.

geometric sequence formula

The core of solving geometric sequences lies within the **geometric sequence formula**. This formula helps calculate the sum of n terms quickly and efficiently. The formula is:

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

Here, \( a \) represents the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula compiles all sequential terms into a singular, simple equation. It enables you to compute the sum faster than adding each term by hand. This organized method is particularly useful for large sequences, saving time and effort.

first term

The first term, usually denoted as \( a \), kicks off the entire sequence. It is the starting point from which the sequence expands. In the exercise provided, \( a = -4 \). This first term is multiplied by successively larger powers of the common ratio to generate the terms of the sequence. Knowing the first term is crucial, because it sets the stage for calculating the entire series, including the sum of all terms up to a given point.

number of terms

The **number of terms** in a geometric sequence, often represented by \( n \), tells you how many items there are from the first term up to the nth one. This number is significant when using the sum formula, as it determines the breadth of the sequence considered. In our example, \( n = 5 \), meaning we sum the first five terms only. By plugging this value into the formula, we can precisely calculate the sum of this particular segment of the sequence.