Question

Find a formula \(a_{n}\) for the \(n\) th term of the geometric sequence whose first term is \(a_{1}=3\) such that \(\frac{a_{n+1}}{a_{n}}=1 / 10\) for \(n \geq 1\).

Short answer

\(a_n = 3 \cdot \left(\frac{1}{10}\right)^{n-1}\)

Step-by-step solution

Understanding the Geometric Sequence

A geometric sequence is defined by its first term and a common ratio. Here, the first term of the sequence is given as \(a_1 = 3\). The common ratio \(r\) is the ratio between consecutive terms, and it is provided as \(\frac{1}{10}\).

Utilizing the Formula for the n-th Term

The general formula for the n-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n-1}\). This formula uses the first term \(a_1\) and the common ratio \(r\), raised to the power of \(n-1\).

Substituting Known Values

Substitute the known values into the formula. Here, \(a_1 = 3\) and \(r = \frac{1}{10}\). Thus, the formula becomes \(a_n = 3 \cdot \left(\frac{1}{10}\right)^{n-1}\).

Simplifying the Expression

The n-th term expression \(a_n = 3 \cdot \left(\frac{1}{10}\right)^{n-1}\) is already simplified. This expression represents the n-th term in the sequence, calculated using the initial term and the common ratio raised to \(n-1\).

Key concepts

Common Ratio

The concept of a "common ratio" is central to understanding a geometric sequence. In any geometric sequence, this ratio is what you multiply a term by to get to the next term in the sequence. For example, if you start with the number 3 and your common ratio is 2, the next number would be 3 times 2, which is 6, then 6 times 2, which is 12, and so on. In our given exercise, the common ratio is much smaller, specifically \(\frac{1}{10}\). This indicates that each term in the sequence will be one-tenth of the previous term, making the numbers decrease quickly.

• The first term in our sequence is given, 3.
• The common ratio is understood to be \(\frac{1}{10}\).
• This means the sequence rapidly approaches zero with each term.

Thus, by understanding the common ratio, you can determine the direction and the rate of change between each consecutive term in your sequence.

n-th Term Formula

The formula for the n-th term of a geometric sequence is a simple yet powerful tool. It is written as \(a_n = a_1 \cdot r^{n-1}\), where:

• \(a_n\) is the n-th term you want to find.
• \(a_1\) is the first term of the sequence, which serves as the starting point.
• \(r\) is the common ratio between successive terms.
• \(n\) is the term number you are trying to calculate.

This formula uses the common ratio and the position in the sequence to calculate the desired term directly. It's especially handy because you don't have to calculate all the previous terms to find the specific term you want.In our exercise, we've calculated \(a_n = 3 \cdot \left(\frac{1}{10}\right)^{n-1}\), which explicitly expresses how to find any term in this specific sequence directly from the formula.

Sequence

A "sequence" in mathematics is simply an ordered list of numbers. For geometric sequences, the numbers are related through multiplication by a common ratio. Each sequence has its own unique pattern, dictated by its first term and common ratio. Given the first term \(a_1 = 3\) and common ratio \(\frac{1}{10}\), we can establish a pattern:

• The first term is 3.
• The second term is \(3 \cdot \frac{1}{10} = 0.3\).
• The third term is \(0.3 \cdot \frac{1}{10} = 0.03\).

As you see, each term is one-tenth of the previous term due to the common ratio \(\frac{1}{10}\). This generates a descending sequence where each number consistently gets smaller, illustrating an exponential decay pattern.Understanding how the "sequence" works provides insight into the behavior of geometric sequences governed by their common ratio and starting term.