Problem 23 - Precalculus: Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}.\) $$a_{n}=2(3)^{n}$$

Step-by-Step Solution

Answer

Yes, it is geometric. The common ratio is 3, and \(a_{n} = 6 \cdot 3^{n-1}\).

Step 1: Identify Given Formula

The exercise provides the formula for the sequence: \(a_{n} = 2(3)^{n}\). Let us analyze this formula to determine the nature of the sequence.

Step 2: Find the First Five Terms

For \(n = 1\) through \(n = 5\), substitute these values into the formula:- \(a_1 = 2(3)^1 = 6\)- \(a_2 = 2(3)^2 = 18\)- \(a_3 = 2(3)^3 = 54\)- \(a_4 = 2(3)^4 = 162\)- \(a_5 = 2(3)^5 = 486\)These values are the first five terms of the sequence: 6, 18, 54, 162, 486.

Step 3: Check if the Sequence is Geometric

A sequence is geometric if the ratio between successive terms is constant. Let's check the ratio of the consecutive terms:- Ratio between \(a_2\) and \(a_1\): \(\frac{18}{6} = 3\)- Ratio between \(a_3\) and \(a_2\): \(\frac{54}{18} = 3\)- Ratio between \(a_4\) and \(a_3\): \(\frac{162}{54} = 3\)- Ratio between \(a_5\) and \(a_4\): \(\frac{486}{162} = 3\)Since the ratio is constant, the sequence is geometric.

Step 4: Determine the Common Ratio

From Step 3, we identified the common ratio \(r = 3\). This confirms the sequence is geometric.

Step 5: Express the General Term in Standard Form

The standard form for the general term of a geometric sequence is given by \(a_{n} = a r^{n-1}\).In this case, \(a = a_1 = 6\) and \(r = 3\). Substitute these values in:\[a_{n} = 6 \cdot 3^{n-1}\]

Key Concepts

Common Ratio

The common ratio in a geometric sequence is a key concept that defines the sequence's growth pattern. It is the constant factor between consecutive terms. If we take any term in the sequence and divide it by the previous term, we'll get the common ratio. For instance, let's explore how it worked in the given example:
- The second term is 18 and the first term is 6, which gives us: \(\frac{18}{6} = 3\). - Similarly, dividing the third term (54) by the second term (18) also gives us 3.
The constancy of this ratio (in our case, always 3) across all consecutive terms proves the sequence is geometric. Understanding the common ratio is important as it dictates how rapidly the sequence values increase or decrease.

Sequence Formula

A sequence formula is an important mathematical tool as it provides a way to determine the terms of a sequence without necessarily computing each term individually. In a geometric sequence, the formula usually involves an initial term and the common ratio.
For example, the sequence given in the exercise is defined with the formula \(a_{n} = 2(3)^{n}\). Here:

• The number 2 signifies the initial amount, which acts similarly to a "starting point" for the sequence.
• The base of the exponential, 3, is the common ratio illustrating how each successive term is generated by multiplying the previous term by 3.

This formula allows you to quickly find any term in the sequence by plugging in the desired position \(n\). Understanding the components of a sequence formula enables you to easily recognize and describe the behavior of a sequence.

General Term of Sequence

The general term of a sequence helps us calculate any term in the sequence using a formula, especially valuable in lengthy or infinite sequences. For geometric sequences, the standard formula for the general term is \(a_{n} = a r^{n-1}\).
In this formula:

• \(a\) is the first term of the sequence.
• \(r\) is the common ratio, the factor by which each term is multiplied to obtain the next term.
• \(n\) is the term number in the sequence you're calculating.

For the sequence in the exercise, substituting \(a = 6\) and \(r = 3\) into the formula gives us \(a_{n} = 6 \cdot 3^{n-1}\). This indicates that to find any term, you can multiply 6 by 3 raised to the power of one less than the term's position number. By mastering the general term formula, you can comfortably handle any inquiries about the sequence's progression or specific elements.