Problem 46 - Algebra 2

Question

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 17 th term

Step-by-Step Solution

Answer

The 17th term of the given geometric sequence is 1970324836974592.

Step 1: Identify the common ratio

To find the common ratio \(r\), divide any term in the sequence by the preceding term. Here, we'll use the first two terms: \(r = 12/3 = 4\).

Step 2: Use the formula for geometric sequence to find the 17th term

Once we know the first term \(a\) (which is 3) and the common ratio \(r\) (which we found to be 4), we can use the formula for the nth term of a geometric sequence \(T_n = ar^{n-1}\). Substituting \(a = 3\), \(r = 4\), and \(n = 17\) into the formula, we get \(T_{17} = 3 \times (4^{17-1})\)

Step 3: Compute the result

Perform the calculation above to find the 17th term: \(T_{17} = 3 \times (4^{16})\). Using a calculator if required, we find that \(T_{17} = 1970324836974592\)

Key Concepts

Common Ratio

In the world of geometric sequences, the common ratio is a key player. It's the factor that helps you leap from one term in the sequence to the next. To find this secret ingredient, all you need to do is divide any term by the one before it.
For example, if you have a sequence that begins with 3, 12, 48, and 192, you'll find the common ratio by taking the second term and dividing it by the first:

• 12 divided by 3 gives you 4.
• Checking further, 48 divided by 12 also equals 4.
• Again, 192 divided by 48 equals 4.

So, in this sequence, the common ratio is 4, meaning each term is four times the size of the previous one.
The beauty of the common ratio is its consistency. No matter where you are in the sequence, you multiply the current term by the common ratio to find the next term.

Nth Term Formula

The nth term formula in a geometric sequence allows you to find any term in the sequence without needing to list all previous terms.
This is magical if you're looking for a far-off term like the 100th or even the 17th, as in this example. The formula is simple and elegant:

• The first term of your sequence is denoted by \( a \).
• The common ratio is \( r \).
• And \( n \) is the term's position in the sequence.

The formula then comes together as \( T_n = ar^{n-1} \).
For the given sequence with \( a = 3 \) and \( r = 4 \), the 17th term \( T_{17} \) is calculated like this:

• Plug in the numbers: \( T_{17} = 3 \times (4^{17-1}) \).
• It simplifies to \( 3 \times (4^{16}) \).

This formula helps you jump directly to the desired term, saving time and effort when dealing with large sequences.

Exponential Growth

A hallmark of geometric sequences is their exponential growth. Unlike arithmetic sequences, which increase by a fixed number, geometric sequences grow by multiplication.
This can lead to dramatic increases as you progress through the sequence. With a constant multiplication by the common ratio, each term gets exponentially larger, showcasing exponential growth.
In the sequence starting with 3, 12, 48, and 192, you notice this growth clearly:

• The second term is 4 times the first.
• The third is 4 times the second.
• Continuing this pattern rapidly escalates the values.

By the time you reach the 17th term, as calculated, you're dealing with an astronomical number like 1970324836974592. This is the power of exponential growth in geometric sequences. It allows quantities to increase rapidly, often explaining patterns seen in areas such as finance, population dynamics, and even computer science.