Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=(-1)^{n}(3 n-2)\\\ &a_{25}= \end{aligned}$$

Step-by-Step Solution

Answer

So, the 25th term of the sequence, \(a_{25}\), is -73.

Step 1: Understand the Given Sequence

The given sequence is \(a_{n}=(-1)^{n}(3 n-2)\). Each term in the sequence can be found by substituting different values of 'n' into the equation.

Step 2: Substitute the value of n

We need to find the 25th term, so we substitute n=25 into the equation. So, our new equation becomes \(a_{25}=(-1)^{25}(3*25-2)\).

Step 3: Solve for \(a_{25}\)

Now we perform the operations to solve for \(a_{25}\). First solve inside the brackets. 3*25 equals 75, from which we subtract 2 to get 73. Also, \( (-1)^{25} = -1 \) because any odd power of negative one is negative. So, our final equation to solve becomes \(a_{25} = -1*73 = -73\).

Key Concepts

Sequence Terms

A sequence in mathematics is essentially a list of numbers arranged in a specific order. Each number in a sequence is known as a "term". In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is referred to as the "common difference".
To find a specific term in any sequence, like the 25th term from our problem, you insert the term number into a general formula. The given sequence formula is \( a_{n} = (-1)^{n}(3n-2) \). By inputting \( n = 25 \), it becomes clear how various components interact to produce each term.
The process of determining a term is simple:

• Insert the term number into the sequence equation.
• Calculate any internal mathematical operations as needed.

This means the terms in this particular sequence alternate between positive and negative values, lending an intriguing pattern to further explore.

Alternating Series

An alternating series is a number series where the terms switch signs with each successive addition or subtraction. This means the terms in the series alternate between positive and negative values.
The sequence formula provided, \(a_{n} = (-1)^{n}(3n-2)\), helps demonstrate this concept. The term \((-1)^{n}\) is the part responsible for the alternating sign. When \(n\) is odd, \((-1)^{n}\) results in \(-1\), giving the term a negative sign. Conversely, when \(n\) is even, \((-1)^{n}\) gives a \(+1\), making the term positive.
This simple alternation of signs can drastically alter the sequence's behavior or its summation. Alternating series are crucial when exploring convergence in calculus or summing series, with considerations of how their behavior affects the sequence or total.

Odd and Even Powers

Odd and even powers are a fundamental mathematical concept involving exponents. When raising numbers to a power, the outcome depends heavily on whether the power is odd or even. When a negative number, like -1, is raised to an odd power, the result remains negative. Conversely, if raised to an even power, it becomes positive.
This principle is crucial in sequences like our example \(a_{n} = (-1)^{n}(3n-2)\). Here, the \((-1)^{n}\) determines the sign of each term by evaluating \(n\). When \(n = 25\) (an odd number), \((-1)^{25}\) equals \(-1\), producing a negative term. This switching of signs based on exponent parity is a powerful tool in predicting the behavior of sequences in various applications.
Understanding how powers function—especially in determining positives and negatives—equips you with the knowledge to tackle diverse mathematical problems and sequences.