Problem 33 - Precalculus with Limits

Question

In Exercises 33-36, find the indicated term of the sequence. \( a_n = (-1)^n (3n - 2) \) \( a_{25} = \Box \)

Step-by-Step Solution

Answer

-73

Step 1: Identify the type of the sequence

The exercise deals with an alternating arithmetic sequence. An important feature of this sequence is that the sign of each term is determined by \((-1)^n\), turning even terms positive and odd terms negative.

Step 2: Substitute the given value of n

Substitute \( n = 25 \) into \( a_n \) to find the required term. That is, solve \( a_{25} = (-1)^{25} (3*{25} - 2) \).

Step 3: Compute the operation

Due to the property of the powers of -1, \((-1)^{25}\) is -1 (because 25 is odd), and \(3*{25} - 2\) is 73. Therefore, the 25th term (denoted as \(a_{25}\)) is -1 * 73 = -73.

Key Concepts

Sequence Term Calculation

When tackling problems involving sequences, one crucial skill is sequence term calculation. This essentially means finding a specific term in a sequence by plugging in the proper values. In the given problem, you have a defined formula for the sequence:

• \( a_n = (-1)^n (3n - 2) \)

Your task is to find the specific term when \( n = 25 \). This is a straightforward substitution process where you insert 25 in place of \( n \) into the formula. This gives:

• \( a_{25} = (-1)^{25} (3\times25 - 2) \).

Following this, all that remains is to simplify the expression for \( n = 25 \). Thus, sequence term calculation becomes an easy step-by-step substitution and simplification task.

Powers of -1

A significant aspect of this exercise involves understanding the powers of -1. Recognizing patterns in these powers is key, especially since they dictate the sign of each term. Here's a brief overview:

• When raising -1 to an even power, such as \((-1)^2\) or \((-1)^4\), you get 1.
• When raising -1 to an odd power, such as \((-1)^1\) or \((-1)^3\), you get -1.

In our exercise, \((-1)^{25}\) appears. Since 25 is an odd number, \((-1)^{25}\) equals -1. This helps determine the sign of the 25th term in our sequence. Understanding these foundational concepts ensures you can swiftly identify the sign of any term within an alternating sequence.

Substitution in Sequences

Substitution in sequences involves replacing a general variable, often \( n \), with a specific number to find a particular term. In this problem, you substitute \( n = 25 \) into the sequence formula:

• \( a_n = (-1)^n (3n - 2) \)

When you substitute \( n = 25 \), you carry out:

• First calculate the exponent of -1, which is \((-1)^{25}\).
• Then, compute \( 3 \times 25 - 2 = 75 - 2 = 73 \).
• Finally, you multiply the result by the sign obtained from \((-1)^{25}\), which leads to \(-1 \times 73 = -73 \).

Substitution is a powerful tool, allowing you to break down complex formulae step-by-step, making calculation manageable for any sequence.