Problem 90 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n} .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Step-by-Step Solution

Answer

Each ratio is equal to 5. The given sequence is a geometric sequence with a common ratio of 5.

Step 1: Calculate the ratios of successive terms

First, calculate the ratios \(\frac{a_{2}}{a_{1}}\), \(\frac{a_{3}}{a_{2}}\), \(\frac{a_{4}}{a_{3}}\), and \(\frac{a_{5}}{a_{4}}\). Substitute \(a_{2}=3 \cdot 5^{2}\), \(a_{1}=3 \cdot 5^{1}\), \(a_{3}=3 \cdot 5^{3}\), \(a_{2}=3 \cdot 5^{2}\), \(a_{4}=3 \cdot 5^{4}\), \(a_{3}=3 \cdot 5^{3}\), \(a_{5}=3 \cdot 5^{5}\), and \(a_{4}=3 \cdot 5^{4}\) into the ratios respectively.

Step 2: Simplify each ratio

Now simplify each ratio by canceling the like terms. The 3's will cancel out and the terms of power will subtract, which will leave just a 5 as a result for every ratio.

Step 3: Identify the observed pattern

The value of each ratio is the same, which is 5. This means that the given sequence is a geometric sequence.

Key Concepts

Sequence Terms

In the world of mathematics, a sequence is an ordered list of numbers following a particular rule. The "terms" of a sequence are the individual elements that are part of this list. For a sequence such as \(a_n = 3 \cdot 5^n\), each term can be determined by substituting different values of \(n\).
For example:

• For \(n = 1\), the first term \(a_1 = 3 \cdot 5^1 = 15\).
• For \(n = 2\), the second term \(a_2 = 3 \cdot 5^2 = 75\).
• Similarly, \(a_3 = 3 \cdot 5^3 = 375\) and so on.

The sequence goes on indefinitely. Recognizing each term in a sequence helps in understanding its overall structure and behavior.

Ratio of Successive Terms

A critical feature of a sequence is the ratio between consecutive, or successive, terms. This can reveal a lot about the sequence's nature. By calculating the ratio of successive terms in a sequence \(a_n = 3 \cdot 5^n\), we aim to determine if there is a consistent pattern.
For example:

• The ratio \(\frac{a_2}{a_1} = \frac{75}{15} = 5\).
• Similarly, \(\frac{a_3}{a_2} = \frac{375}{75} = 5\).
• The ratio \(\frac{a_4}{a_3} = 5\) and \(\frac{a_5}{a_4} = 5\) also yield the same value.

The consistency in these ratios indicates the sequence is geometric, characterized by each term being multiplied by a constant ratio, often referred to as the common ratio.

Powers of Numbers

In sequences, terms often involve powers of numbers, which significantly affect their magnitude and pattern. For our sequence, \(a_n = 3 \cdot 5^n\), powers play a crucial role as each term is a result of 5 raised to the power of \(n\).
Consider these factors:

• Raising a number to a higher power increases its value exponentially.
• The base number (5 in this case) dictates the growth rate of terms.
• Understanding powers helps solve sequences by simplifying calculations like those of successive terms.

Using powers provides insight into how quickly sequences can grow or shrink, especially in terms that involve exponential functions.

Mathematical Patterns

Mathematical patterns are all around us, especially within sequences. A sequence's pattern can be identified by looking at how terms relate to each other. In a geometric sequence, the pattern is determined by the common ratio, which stays constant between successive terms.
Identifying patterns is crucial because:

• They simplify complex mathematical relationships into manageable forms.
• Patterns help in predicting future terms and understanding overall behavior.
• In a geometric sequence, knowing the pattern allows for the calculation of any term without having to list all preceding terms.

Recognizing and understanding these patterns can greatly aid in solving a variety of mathematical problems, making them an essential part of mathematical learning.