Problem 59 - Algebra 2

Question

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=1, r=5 $$

Step-by-Step Solution

Answer

The explicit formula for the geometric sequence is \( a_{n} = 5^{(n-1)} \). The first three terms of the sequence are `1`, `5`, and `25`.

Step 1: Writing the Explicit Formula

To write the explicit formula, replace \( a_{1} \) and \( r \) in the formula \( a_{n} = a_{1} \cdot r^{(n-1)} \) with the given values. So, for \( a_{1} = 1 \) and \( r = 5 \), the explicit formula will be \( a_{n} = 1 \cdot 5^{(n-1)} \). This simplifies to \( a_{n} = 5^{(n-1)} \).

Step 2: Generating the First Term

To find the first term in the sequence, substitute `n=1` into the explicit formula. This gives: \( a_{1} = 5^{(1-1)} = 5^0 = 1 \). So, the first term of the sequence is `1`.

Step 3: Generating the Second Term

To find the second term, substitute `n=2` into our explicit formula. Therefore, we get \( a_{2} = 5^{(2-1)} = 5^1 = 5 \). So, the second term is `5`.

Step 4: Generating the Third Term

To find the third term, substitute `n=3` into the explicit formula. This gives \( a_{3} = 5^{(3-1)} = 5^2 = 25 \). So, the third term in the sequence is `25`.

Key Concepts

Explicit Formula

In the realm of geometric sequences, the explicit formula is a powerful tool that allows you to determine any term in the sequence without calculating all previous ones. The general form of the explicit formula for a geometric sequence is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Here, \( a_1 \) is the first term in the sequence, \( r \) is the common ratio, and \( n \) represents the term number you wish to find. By plugging in these values, you can find any term directly.
This means by knowing just the first term and the common ratio, you can find any term in a blink. For example, in the sequence where \( a_1 = 1 \) and \( r = 5 \), the explicit formula becomes:

• \( a_n = 1 \cdot 5^{(n-1)} \)
• This simplifies to: \( a_n = 5^{(n-1)} \)

Knowing how to set up this formula is crucial for efficiently working with geometric sequences.

Sequence Terms

Sequence terms in a geometric sequence are generated from the explicit formula by substituting the position number of the term. Each term builds on the previous term's multiplication by the common ratio.
To illustrate, let's consider the sequence with \( a_1 = 1 \) and \( r = 5 \):

• First term: Substitute \( n=1 \): \( a_1 = 5^{(1-1)} = 5^0 = 1 \)
• Second term: Substitute \( n=2 \): \( a_2 = 5^{(2-1)} = 5^1 = 5 \)
• Third term: Substitute \( n=3 \): \( a_3 = 5^{(3-1)} = 5^2 = 25 \)

Each term follows sequentially by multiplying the previous term by the common ratio, showing how a single pattern governs the entire sequence.

Common Ratio

The common ratio \( r \) is a fundamental feature of geometric sequences that dictates how the sequence progresses. It is the constant factor by which each term is multiplied to get the next term.
In our example:

• \( r = 5 \) means each term is five times the previous term.
• This constant ratio allows the sequence to "grow" or "shrink" exponentially, depending on whether \( r \) is greater than one or a fraction.

Understanding the common ratio emphasizes the exponential nature of changes in geometric sequences:

• If \( r = 1 \), the sequence stays constant.
• If \( r > 1 \), the sequence increases rapidly.
• If \( 0 < r < 1 \), the sequence decreases towards zero.

Always determining this ratio is key to deriving any other term and understanding the overall behavior of the sequence.