Problem 34 - Precalculus: Mathematics for Calculus

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(3,0.3,0.03,0.003, \dots\)

Step-by-Step Solution

Answer

The nth term is given by \(a_n = 3 \times (0.1)^{(n-1)}\).

Step 1: Identify the Pattern

Observe the sequence: 3, 0.3, 0.03, 0.003, .... Notice that each term is obtained by multiplying the previous term by 0.1.

Step 2: Recognize the Sequence Type

The sequence is a geometric sequence because each term is a fixed ratio (0.1) times the previous term. In a geometric sequence, each term is given by multiplying the previous term by the common ratio.

Step 3: Write the General Term of the Sequence

The general form of a geometric sequence is given by the formula \(a_n = a_1 imes r^{(n-1)}\), where \(a_1\) is the first term (3 in this case), \(r\) is the common ratio (0.1), and \(n\) is the term number.

Step 4: Substitute the Known Values into the Formula

Substitute \(a_1 = 3\), \(r = 0.1\), into the formula to get \(a_n = 3 imes (0.1)^{(n-1)}\).

Step 5: Simplify the Formula

The formula \(a_n = 3 imes (0.1)^{(n-1)}\) simplifies directly to provide the formula for the nth term of the sequence.

Key Concepts

Common Ratio

In a geometric sequence, it is crucial to identify the common ratio as it dictates how the sequence progresses. The common ratio is the constant factor that each term is multiplied by to produce the next term.
For instance, in the sequence provided: 3, 0.3, 0.03, 0.003, ..., each term is obtained by multiplying the previous one by 0.1.

• If you divide any term by the previous term, you will find the common ratio, which in this sequence is 0.1.
• This value remains consistent throughout the sequence.

Identifying the common ratio ensures the recognition of the sequence as geometric, setting the foundation for further calculations.

Nth Term Formula

The nth term formula in a geometric sequence helps determine the value of any term in the sequence without having to list all previous terms.
The formula is expressed as: \[ a_n = a_1 \times r^{(n-1)} \] Where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term in the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

For the sequence provided, the formula allows us to replace \( a_1 \) with 3 and \( r \) with 0.1. Plugging these into the formula gives \[ a_n = 3 \times (0.1)^{(n-1)} \].
This formula lets you efficiently calculate any term in the sequence without any guesswork.

Sequence Pattern

Recognizing the pattern in a sequence is key to understanding its nature. The pattern tells us whether a sequence is arithmetic, geometric, or another type.
In this exercise, we see the sequence starts at 3 and each subsequent term is formed by multiplying the previous one by 0.1.

• This regular multiplication by a constant factor confirms it's a geometric sequence, dictated by a predictable pattern.
• Observing this multiplication consistency helps us set the general formula to apply.

Knowing the pattern allows you to predict future terms and understand the sequence's behavior.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific rule. These sequences can be arithmetic, geometric, or any other type, which depicts how terms relate to each other.
A geometric sequence like the one in this exercise has its terms derived by multiplying a constant, the common ratio, to the previous term.

• Understanding sequences involves recognizing this rule or pattern between terms.
• Sequences are used in various fields of math and science to model and predict behaviors.

By applying these rules systematically, sequences help solve complex problems efficiently.