Problem 21 - College Algebra

Question

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$1.5,-3,6,-12, \dots$$

Step-by-Step Solution

Answer

The seventh term of the given geometric sequence is 96

Step 1: Find the common ratio

The common ratio for a geometric sequence is derived by dividing any term by its predecessor. Given the sequence \(1.5,-3,6,-12, \dots\), we can find the common ratio \(r\) by dividing -3 by 1.5 or 6 by -3 or -12 by 6. All these yield a common ratio \(r\) of -2.

Step 2: Find general formula for \(a_{n}\)

Next, using the formula for the general term of a geometric sequence, we get \(a_{n} = a_{1} * r^{n-1}\), where \(a_{1}\) is the first term in the sequence and \(r\) is the common ratio of the sequence. Here \(a_{1} = 1.5\) and \(r = -2\). So our general term is \(a_{n} = 1.5 * (-2)^{n-1}\)

Step 3: Find the seventh term

To find the seventh term \(a_{7}\), substitute \(n = 7\) into the formula and calculate: \(a_{7} = 1.5 * (-2)^{7-1} = 1.5 * (-2)^{6} = 1.5 * 64 = 96\)

Key Concepts

Understanding the Common Ratio

A geometric sequence is defined by a sequence of numbers in which each term is a constant multiple of the previous one. This constant multiple is known as the common ratio, often denoted as r. In essence, to find the common ratio of a geometric sequence, you simply divide one term by the term before it.

For example, given a sequence like 1.5, -3, 6, -12,..., calculating the ratio between -3 and 1.5 yields -2. Similarly, dividing 6 by -3 or -12 by 6 also gives us the same common ratio of -2. This indicates that each term is -2 times its preceding term, adhering to the defining property of a geometric sequence.

The common ratio is vital because it’s the backbone of the sequence’s structure, dictating how the sequence will progress. It's what enables us to predict future terms and establish a formula for any term in the sequence.

Finding the nth Term of a Geometric Sequence

Once you know the common ratio, finding each subsequent term within a geometric sequence becomes straightforward using the nth term formula. The nth term, often represented as a_n, of a geometric sequence can be found using the formula: \[a_n = a_1 \times r^{(n-1)}\] where a₁ is the first term, and n is the term number you want to find.

By inserting the known values into the formula, you can calculate any term's value. For instance, with a first term of 1.5 and a common ratio of -2, you could find the seventh term in the sequence by substituting these values into the nth term formula:
\[a_7 = 1.5 \times (-2)^{(7-1)}\] This process highlights the power of the formula, allowing an exact term in the sequence to be pinpointed without needing to list out all terms up to that point.

Applying the Geometric Sequence Formula

When approaching problems involving geometric sequences, understanding how to apply the geometric sequence formula is imperative. It's a potent tool that enables students to map out the entire sequence based on its initial term and common ratio.

Let's revisit our example – the sequence beginning with 1.5 and continuing with a common ratio of -2. The general formula for our sequence is: \[a_n = 1.5 \times (-2)^{(n-1)}\]

To leverage this formula effectively, remember that the exponent reflects how many times you need to multiply the common ratio by the sequence's first term. A common pitfall is to confuse the term's index with the exponent applied to the common ratio, but always remember that our exponent is n-1, ensuring we multiply the correct number of times.

Calculating the seventh term gives us: \[a_7 = 1.5 \times (-2)^{6} = 96\] This optimized approach makes it possible to solve for any term, without repetitively multiplying terms or employing laborious calculations. It’s a clear demonstration of the elegance and efficiency that mathematics provides to solve what might otherwise be a daunting sequence to decipher. Understanding and applying this formula will undoubtedly solidify students' grasp on geometric sequences, allowing them to tackle more complex sequences and mathematical problems with confidence.