Problem 32 - Algebra for College Students

Question

Find the sum of the first nine terms of the geometric sequence \(-2,6,-18,54, \ldots .\) \(-9842\)

Step-by-Step Solution

Answer

The sum of the first nine terms is -9842.

Step 1: Identify the first term and common ratio

The first term of the sequence is given as \(a_1 = -2\). The common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{6}{-2} = -3\). This means each term is multiplied by \(-3\) to get to the next term.

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

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by the formula \[ S_n = a_1 \frac{1-r^n}{1-r} \] where \(a_1\) is the first term and \(r\) is the common ratio.

Step 3: Substitute the known values into the formula

Substitute \(a_1 = -2\), \(r = -3\), and \(n = 9\) into the formula: \[ S_9 = -2 \frac{1-(-3)^9}{1-(-3)} \].

Step 4: Compute \((-3)^9\) and simplify

Calculate \((-3)^9\): \[ (-3)^9 = -19683 \]. Substitute back into the formula: \[ S_9 = -2 \frac{1 - (-19683)}{1 + 3} = -2 \frac{1 + 19683}{4} \].

Step 5: Compute the expression and simplify further

Evaluate \(1 + 19683 = 19684\). Substitute this into the expression: \[ S_9 = -2 \frac{19684}{4} \mathrm{which} \mathrm{is} -2 \times 4921 \].

Step 6: Calculate the final sum

Finally, multiply: \[ -2 \times 4921 = -9842 \].Thus, the sum of the first nine terms of the geometric sequence is \(-9842\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each term is derived by multiplying the previous term by a fixed number. This fixed number is called the common ratio. In our specific example, we are asked to find the sum of the first nine terms of the sequence \(-2, 6, -18, 54, \ldots\). Here, the first term is \(a_1 = -2\). To find the common ratio \(r\), we divide the second term by the first term: \(r = \frac{6}{-2} = -3\). This reveals that if you multiply any term by \(-3\), you'll get the next term. By identifying the common ratio, we can quickly predict subsequent terms in the sequence. For example, multiplying \(-18\) by \(-3\) results in \(54\). This step is key to understanding and working with geometric sequences, as it lets us use the same straightforward rule to find any term.

Calculating the Sum of a Series

When dealing with geometric sequences, calculating the sum of a specific number of terms involves using a formula. The formula to find the sum of the first \(n\) terms \(S_n\) of a geometric sequence is:\[ S_n = a_1 \frac{1-r^n}{1-r} \]Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms to sum. This formula is especially helpful, as directly adding all terms can be time-consuming when \(n\) is large. In our example, we need the sum of the first nine terms. We substitute our known values into the formula: \(a_1 = -2\), \(r = -3\), and \(n = 9\). Careful substitution into the formula gives:\[ S_9 = -2 \frac{1-(-3)^9}{1-(-3)} \] This helps break down the task of summing difficult sequences into manageable arithmetic steps that we can easily compute.

Performing Arithmetic Operations in the Formula

Using the formula for geometric sequences involves performing specific arithmetic operations, including exponentiation and division. Let's delve into these steps using our example:1. **Calculate the power of the common ratio:** Compute \((-3)^9\). As \((-3)^9 = -19683\), it illustrates the impact of repeatedly multiplying the common ratio.2. **Substitute back:** Insert this value back into the formula: \[ S_9 = -2 \frac{1 - (-19683)}{1 + 3} \]3. **Simplify and calculate:** Simplify the fraction \(1 + 19683 = 19684\), and divide by \(4\):\[ \frac{19684}{4} = 4921 \]4. **Final multiplication:** Finally, multiply by \(-2\):\[ -2 \times 4921 = -9842 \]Each operation, whether adding, multiplying, or simplifying, is essential to determining the series' sum. Precision in these arithmetic steps is crucial for the correct result.