Problem 17 - Algebra and Trigonometry

Question

In Exercises \(17-24,\) 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. $$ 3,12,48,192, \dots $$

Step-by-Step Solution

Answer

The seventh term of the given geometric sequence is 12288.

Step 1: Identify the first term and the ratio

The first term \(a_{1}\) of the given sequence is 3. To find the common ratio, divide the second term by the first term, or the third term by the second term and so on. Doing this we find that \(r = 12 / 3 = 4\). So, the first term a_{1} is 3 and the common ratio \(r\) is 4.

Step 2: Formulate the formula for the nth term of the geometric sequence

The formula for the nth term of a geometric sequence is \(a_{n} = a_{1} * r^{(n-1)}\). Substituting our found values in, we find the formula for the nth term for this sequence is \(a_{n} = 3 * 4^{(n-1)}\).

Step 3: Use the formula to find the seventh term of the sequence

Now, we can use the derived formula to find the seventh term of the sequence. Substitute \(n=7\) into the formula, we get \(a_{7} = 3 * 4^{(7-1)} = 3 * 4096 = 12288\).

Key Concepts

nth term

In a geometric sequence, the nth term is all about understanding the position and value of a particular element within the sequence. It is important because it helps us predict any term in the sequence without having to list all of them.
The formula for the nth term of a geometric sequence is given by: \( a_{n} = a_{1} \cdot r^{(n-1)} \).

• \(a_{n}\) represents the nth term of the sequence.
• \(a_{1}\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number.

When you substitute the known values for \(a_{1}\) and \(r\), while also choosing a specific \(n\), you can easily calculate any nth term.

common ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It's crucial to identifying the pattern of the sequence.
To find the common ratio \(r\), you divide any term in the sequence by the previous term. For instance, using the sequence provided:

• Divide the second term by the first term, \(\frac{12}{3} = 4\).
• You can also divide the third term by the second term, \(\frac{48}{12} = 4\).

Either way, you will determine that the common ratio \(r\) is 4. Understanding the common ratio is essential because it assists not only in forming the nth term formula but also in predicting the behavior of the sequence.

seventh term calculation

Calculating the seventh term effectively involves plugging into the nth term formula. Once you've established the formula, the process becomes straightforward.
Given the formula \[ a_{n} = 3 \times 4^{(n-1)} \], you need to substitute \(n = 7\) to find the seventh term.
This looks like:

• Calculate \(4^{(7-1)} = 4^6 = 4096\). You raise 4 to the power of 6.
• Multiply the result by the first term, \(3 \times 4096\), giving you \(12288\).

Therefore, the seventh term \(a_{7}\) of this geometric sequence is 12288. This calculation highlights how understanding and using the formula can make working with sequences much simpler.

general term formula

The general term formula is a powerful mathematical expression that provides a universal way to find any term in a sequence, given the position of the term required. In a geometric sequence, this formula is: \( a_{n} = a_{1} * r^{(n-1)} \).
Breaking down its components:

• \(a_{1}\), the first term, anchors the sequence.
• The exponent \(r^{(n-1)}\) involves the common ratio raised to one less than the term number, reflecting the repeated multiplication needed to reach \(a_{n}\).

To derive the general term for the given sequence of 3, 12, 48, 192, we fill in \(a_{1} = 3\) and \(r = 4\), yielding \[ a_{n} = 3 \times 4^{(n-1)} \].Utilizing this formula effectively lets us swiftly find any desired term, illustrating the sequence pattern elegantly. This clarity and efficiency is why having a general term formula is so important.