Problem 26 - Algebra and Trigonometry

Question

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 nth term is given by \(a_n = 5 \times (-5)^{n-1}\).

Step 1: Identify the Pattern

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

Step 2: Determine the First Term and Common Ratio

The first term of the sequence, denoted as \(a_1\), is 5. The common ratio \(r\) is the factor by which we multiply to get to the next term, which is -5.

Step 3: Write the General Formula for the nth Term

For a geometric sequence, the formula for the \(n\)th term \(a_n\) is given by \(a_n = a_1 \cdot r^{n-1}\). Here, \(a_1 = 5\) and \(r = -5\), so the formula becomes: \(a_n = 5 \times (-5)^{n-1}\).

Step 4: Confirm the Formula with Given Terms

Verify that the formula \(a_n = 5 \times (-5)^{n-1}\) works for the given terms in the sequence. For example:- \(n=1\): \(a_1 = 5\times (-5)^{0} = 5\)- \(n=2\): \(a_2 = 5\times (-5)^{1} = -25\)- \(n=3\): \(a_3 = 5\times (-5)^{2} = 125\)All values match the terms given in the sequence.

Key Concepts

nth term formula

Understanding the formula for the nth term in a geometric sequence is essential. This formula allows you to find any term in the sequence without listing all preceding terms. For any geometric sequence, the nth term can be calculated using the formula: \[ a_n = a_1 \cdot r^{n-1} \] where:

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

This means you start with the first term and multiply by the common ratio raised to a power that is one less than the position of the term in the sequence. For example, if you want to find the 4th term, you would use \(a_1\) multiplied by \(r^{3}\). This formula is efficient and saves you from doing repetitive multiplication to find each term.

common ratio

In a geometric sequence, the common ratio is the constant factor by which each term is multiplied to get the next term.
The common ratio is key to identifying and working with geometric sequences.
To find the common ratio, divide any term by its preceding term. In our example sequence: 5, -25, 125, -625, the common ratio \(r\) is calculated as:

• \(-25 \div 5 = -5\)

Knowing the common ratio allows you to predict future terms and create the formula for the nth term. A negative common ratio will alternate the signs of the terms in the sequence, as seen in our example. Remember, the common ratio can greatly affect the behavior of the sequence.

sequence pattern

Identifying the sequence pattern is the first step in understanding any sequence. For a geometric sequence, the pattern is established by recognizing a consistent ratio between consecutive terms.
In our sequence: 5, -25, 125, -625, we noticed each term was derived by multiplying the previous term by -5.

• This pattern of multiplication by a consistent factor is what categorizes a sequence as geometric.

The pattern also alternates in signs due to the negative common ratio, producing alternating positive and negative terms. Being able to spot these patterns helps not just in solving problems related to sequences, but also in determining the relationship between terms, which is fundamental in many areas of mathematics. Pattern recognition is a critical skill that extends beyond math into analytical thinking in real-world scenarios.