Problem 28 - Algebra and Trigonometry

Question

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 5,-25,125,-625, \dots $$

Step-by-Step Solution

Answer

The \(n\)th term of the sequence is \(a_n = 5 \times (-5)^{n-1}\).

Step 1: Identify the pattern

Observe the given sequence: 5, -25, 125, -625,... Notice that each term is obtained by multiplying the previous term by -5. This indicates the sequence is geometric with a common ratio of -5.

Step 2: Find the first term and common ratio

First term (\(a\)) is 5. Common ratio (\(r\)) is -5, as each term is 5 times the previous term but with a change in sign.

Step 3: Use the general formula for the \(n\) th term of a geometric sequence

The general formula for the \(n\) th term \(a_n\) of a geometric sequence is given by: \[ a_n = a imes r^{n-1} \]Where \(a\) is the first term and \(r\) is the common ratio.

Step 4: Substitute the values into the formula

In the formula, substitute \(a = 5\) and \(r = -5\):\[ a_n = 5 \times (-5)^{n-1} \]

Step 5: Simplify the expression

The expression \(a_n = 5 \times (-5)^{n-1}\) represents the \(n\) th term of the sequence, which includes both the magnitudes and alternating signs as the sequence progresses.

Key Concepts

nth term of a sequence

Understanding the nth term of a sequence is like finding the key to unlock the entire sequence. For a geometric sequence, this term holds particular elegance, as it allows you to determine any term without listing all preceding terms. The nth term is built from the sequence's foundational elements: the first term and the common ratio. For instance, consider the geometric sequence starting with 5 and a pattern of multiplying by -5: 5, -25, 125, -625, ...

To find the nth term, use the general formula:

• First term (\(a\)) = 5
• Common ratio (\(r\)) = -5

With these values:\[a_n = a \times r^{n-1} = 5 \times (-5)^{n-1}.\]This equation not only calculates each specific term's value but mirrors how the sequence naturally evolves with alternating signs.

common ratio

The common ratio in a geometric sequence is the factor by which you multiply the previous term to get the next one. Recognizing this ratio helps you predict how the sequence behaves as it progresses. In our example sequence 5, -25, 125, -625, ..., the common ratio is -5. This implies that each term is produced by multiplying the preceding term by -5, adding a twist of alternating signs in the sequence.

Identifying the common ratio involves a simple division of a term by its preceding term:

• \(-25\) divided by \(5\) equals \(-5\)
• \(125\) divided by \(-25\) equals \(-5\)

This consistent pattern (\(-5\)) highlights the geometric nature of the sequence and serves as a crucial component in defining the nth term formula.

sequence pattern analysis

Sequence pattern analysis is crucial for understanding and predicting long-term behavior of sequences. In our geometric sequence example: 5, -25, 125, -625, ... examining the pattern reveals important characteristics.

Observation tells us:

• Each term alternates in sign due to the common ratio of -5.
• This pattern of alternating signs results from multiplying by a negative number.

Knowing these traits, you can anticipate future terms in the sequence without calculating each one individually. This analysis simplifies the task of determining the nth term for large values of n. Recognizing such patterns helps in understanding the sequence dynamically and enables logical prediction about its future evolution.