Problem 59 - Algebra 2

Question

Find \(a_{1}\) for a geometric sequence with the given terms. $$ a_{5}=112 \text { and } a_{7}=448 $$

Step-by-Step Solution

Answer

The first term \(a_{1}\) of the geometric sequence is 0.4375

Step 1: Find the Ratio

First, calculate the ratio \(r\) of the geometric sequence. The ratio can be found by dividingthe 7th term by the 5th term. So, \(r = a_{7} / a_{5} = 448 / 112 = 4\).

Step 2: Calculate the First Term

After finding the ratio, we can find the first term \(a_1\) of the geometric sequence by using the formula: \(a_{1} = a_{5} / (r^{5-1}) = 112 / (4^4) = 112 / 256 = 0.4375\).

Key Concepts

First Term

In a geometric sequence, the first term is denoted as \(a_1\). The first term is crucial because it sets the initial value of the sequence from which all other terms are derived. To find \(a_1\), we often use the known terms of the sequence along with the sequence's common ratio.
For instance, if you are given the fifth and seventh terms of a sequence and know the common ratio, you can calculate \(a_1\) using the formula:

• \(a_n = a_1 \cdot r^{n-1}\)

Where \(a_n\) is the nth term, \(r\) is the common ratio, and \(n\) is the term number. Solving for \(a_1\) using the parameters given and the ratio leads to the calculation seen in the example.

Common Ratio

The common ratio \(r\) in a geometric sequence is the constant factor between consecutive terms. It is a defining feature of the sequence. For any two consecutive terms \(a_n\) and \(a_{n+1}\), the ratio is calculated as \(r = a_{n+1} / a_n\).
In the example provided, the common ratio is found by dividing the seventh term by the fifth term:

• \(r = a_7 / a_5 = 448 / 112\)
• This simplifies to \(r = 4\)

Finding the common ratio is necessary because it allows you to determine any term in the sequence if you know just one term and the ratio. It's a powerful tool when dealing with growth patterns in sequences.

Exponential Growth

Geometric sequences exemplify exponential growth or decay, depending on the value of the common ratio. When \(r>1\), the sequence grows as each term is a multiple of the previous one by the ratio. Conversely, if \(0In our example, the geometric sequence grows because the common ratio is \(r=4\). This tells us that each term is four times the previous term. Exponential growth in sequences results in very large numbers quickly, which is why the seventh term is much larger than the first term calculated.

• Rapid change over short intervals
• Significant increases with each term

Understanding exponential growth in the context of geometric sequences allows us to grasp how quantities can increase or decrease rapidly in various real-world scenarios.