Problem 26 - Precalculus: Mathematical for Calculus

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 \(a_n = (-1)^{n+1} \times 5^n\).

Step 1: Identify the pattern

Examine the given terms: 5, -25, 125, -625. Notice the alternating signs and that each term's absolute value is a power of 5. Specifically, 5, -5^2, 5^3, -5^4, indicating a pattern of consecutive powers of 5.

Step 2: Determine the general term

Observe that the sign alternates, and the change in sign can be represented by an exponent applied to (-1). Notice that:- The first term (n=1) can be written as \(5^1 = 5\).- The second term (n=2) can be written as \(-5^2 = -25\).- The third term (n=3) can be written as \(5^3 = 125\), and so on.To incorporate the alternating sign, we use \((-1)^{n+1}\). Therefore, the general term \(a_n\) is:\[ a_n = (-1)^{n+1} \times 5^n \]

Step 3: Write the formula for the nth term

Based on the identified pattern, the formula for the nth term of the sequence is:\[ a_n = (-1)^{n+1} \times 5^n \].This captures both the alternating signs and the powers of 5.

Key Concepts

Alternating Sign

When looking at a sequence with terms such as 5, -25, 125, -625, it's easy to notice the alternating signs. Alternating signs mean that each subsequent term switches between positive and negative.
This pattern is often influential in forming the general rule of the sequence. In the given example, the switch in signs can be represented using \((-1)^{n+1}\).
The expression \((-1)^n\) produces

• a positive outcome when \(n\) is even, and
• a negative outcome when \(n\) is odd.

For this sequence, \(n+1\) in the exponent ensures that the sequence starts with a positive number and alternates correctly between positive and negative signs as we check further terms.

Nth Term

In sequences, the goal is often to find a mathematical expression that gives any term's value, often referred to as the nth term. This expression allows you to find any term in the sequence without listing all previous terms.
For example, in the sequence 5, -25, 125, -625, we want to describe a general formula for any term \(a_n\).
According to the given solution, the nth term is \(a_n = (-1)^{n+1} \times 5^n\).

• The \((-1)^{n+1}\) part is responsible for alternating the sign of each term.
• The \(5^n\) component expresses that each term is a power of 5.

Using this formula, we can find the nth term for any integer value of \(n\). For instance, to find the 4th term, substitute \(n = 4\) into the formula to get: \((-1)^{4+1} \times 5^4 = -625\).

Power

Power in mathematics refers to the result of multiplying a number by itself a certain number of times. In the sequence 5, -25, 125, -625, each term (without considering the sign) is a power of 5.
This means:

• The first term is \(5 = 5^1\).
• The second term is \(-25 = -5^2\), which can be interpreted as taking 25 (the second power of 5) but with a sign change.
• The third term is \(125 = 5^3\).

The general rule, \(5^n\) where \(n\) denotes the term's position (like first, second, third), helps in calculating terms.
Thus, for any \(n\)th term in the sequence, \(5^n\) gives us the base value before applying the sign from the alternating pattern. The concept of power allows one to represent repeated multiplication succinctly and is central to expressing the terms of this sequence efficiently.