Problem 70 - Algebra 2

Question

In a geometric sequence, \(a_{1}=3\) and \(a_{4}=192 .\) Explain how to find \(a_{2}\) and \(a_{3} .\)

Step-by-Step Solution

Answer

The second term \(a_2\) is 12 and the third term \(a_3\) is 48

Step 1: Find the Ratio

From the formula \(a_n = a_1*r^{n-1}\), when n=4, we have \(a_4 = a_1*r^3\). We know \(a_1 = 3\) and \(a_4 = 192\), so we can write the equation as: 192 = 3*r^3. Therefore r^3 = 64 and by taking cube root we get r=4.

Step 2: Solve for \(a_2\)

Using the formula \(a_n = a_1*r^{n-1}\) with n=2 and r=4, we get \(a_2 = 3*4^1 = 12\)

Step 3: Solve for \(a_3\)

Now replacing n with 3 in the formula \(a_n = a_1*r^{n-1}\) with r=4, we get \(a_3 = 3*4^2 = 48\)

Key Concepts

Common Ratio

In a geometric sequence, one of the primary features is the 'common ratio'. This ratio helps you move from one term to the next. If you know any term in the sequence and the common ratio, you can find any other term. For example, in the sequence given, it starts with 3 and jumps to 192 by the 4th term. To find the common ratio, we leverage the general formula for any term in a geometric sequence:

• The formula is: \[a_n = a_1 \cdot r^{n-1}\]
• Using the values \(n = 4\), \(a_1 = 3\), and \(a_4 = 192\), we form the equation: \[192 = 3 \cdot r^3\]
• By solving, we find \(r^3 = 64\). Thus, the common ratio \(r = 4\).

The common ratio of 4 indicates the sequence is expanding rapidly, enlarging each term by a factor of four.

Terms of a Sequence

When we refer to the 'terms of a sequence', we are speaking about specific values in the order of numbers defined by the sequence. In a geometric sequence, terms are connected by the common ratio we discussed above.

• Starting with the first term, \(a_1 = 3\), each subsequent term is calculated by multiplying the previous term by the common ratio \(r\).
• For the second term, \(a_2\), we use the formula again: \[a_2 = a_1 \cdot r^{1} = 3 \cdot 4 = 12\]
• Next, for the third term, \(a_3\): \[a_3 = a_1 \cdot r^{2} = 3 \cdot 4^2 = 48\]

The sequence here is exponential, starting with 3 and progressing through 12 and 48 as the second and third terms respectively.

Exponential Growth

Exponential growth in sequences means that values increase rapidly by a constant factor, rather than a fixed amount. This is exactly what happens in a geometric sequence due to its common ratio.

• Each term is a multiple of the previous one by the common ratio, so changes in the sequence are not linear, but rather exponential.
• In this sequence, every term is 4 times the last one, showcasing exponential growth.
• This rapid increase means later terms become significantly larger, highlighting a key characteristic of geometric sequences.

Understanding this property helps in recognizing and predicting patterns in various fields, such as economics and population studies.