Problem 2 - Algebra and Trigonometry

Question

The sequence \(a_{n}=a r^{n-1}\) is a geometric sequence in which a is the first term and \(r\) is the _____ _____. So for the geometric sequence \(a_{n}=2(5)^{n-1}\) the first term is _____ and the common ratio is _____.

Step-by-Step Solution

Answer

First term is 2; common ratio is 5.

Step 1: Identify First Term

In the geometric sequence formula \( a_{n} = a r^{n-1} \), \( a \) represents the first term of the sequence. In the given sequence \( a_{n} = 2(5)^{n-1} \), the value of \( a \) is 2. Thus, the first term is \( a = 2 \).

Step 2: Identify Common Ratio

In the geometric sequence formula \( a_{n} = a r^{n-1} \), \( r \) represents the common ratio. In the given sequence \( a_{n} = 2(5)^{n-1} \), the value of \( r \) is 5. This is the factor by which each term is multiplied to get the next term.

Key Concepts

First Term

In a geometric sequence, the first term is an essential component. It's where the sequence begins, and knowing this term is crucial for generating the other terms. The first term in a sequence is denoted by the symbol \(a\) in the general geometric sequence formula \(a_{n} = a r^{n-1}\). This formula helps you calculate any term in the sequence using just two values: the first term \(a\) and the common ratio \(r\). In the given example, \(a_{n} = 2(5)^{n-1}\), the first term \(a\) is clearly 2. This means the sequence starts with the number 2. Every subsequent term builds on this initial value, making it the foundation of the geometric sequence.

Common Ratio

The common ratio in a geometric sequence is a key factor that dictates how the terms in the sequence grow or shrink. It is represented by the letter \(r\) in the sequence formula \(a_{n} = a r^{n-1}\). The common ratio is the constant factor by which you multiply one term to get to the next term in the sequence.
In our provided example \(a_{n} = 2(5)^{n-1}\), the common ratio \(r\) is 5. This tells us that to move from one term to the next, we simply multiply the current term by 5.

• If the common ratio is greater than 1, the terms in the sequence will increase.
• If it is between 0 and 1, they will decrease.
• If the common ratio is negative, the terms will alternate in sign.

Understanding the common ratio helps in predicting the nature and behavior of the sequence.

Sequence Formula

The sequence formula for a geometric sequence is a powerful tool. It allows for the calculation of any term within the sequence. The formula is expressed as \(a_{n} = a r^{n-1}\), where:

• \(a_{n}\) is the \(n\)-th term of the sequence.
• \(a\) is the first term, setting the starting point of the sequence.
• \(r\) is the common ratio, defining the relationship between consecutive terms.
• \(n\) is the term number, indicating the position of the desired term within the sequence.

To use this formula effectively, you need to know the first term and the common ratio. This information allows you to determine any term in the sequence by substituting the term number \(n\) into the formula. For example, if you wish to find the 3rd term in our specific sequence, substitute \(n=3\) into \(a_{3} = 2(5)^{3-1}\), giving \(a_{3} = 2(5)^{2} = 50\). Consequently, this formula not only helps identify terms but also explains how the sequence is constructed, giving deeper insight into its structure.