Problem 8 - College Algebra

Question

Write the first four terms of the sequence. $$a_{n}=-(-5)^{n-1}$$

Step-by-Step Solution

Answer

The first four terms are -1, 5, -25, 125.

Step 1: Understand the Sequence Formula

The sequence is defined by the formula \(a_n = -(-5)^{n-1}\). This means that for each term \(n\), the term value is \(-1\) times \(-5\) raised to the power \((n-1)\).

Step 2: Calculate the First Term

For the first term, set \(n = 1\). Substitute into the formula: \(a_1 = -(-5)^{1-1} = -(-5)^0 = -1\).

Step 3: Calculate the Second Term

For the second term, set \(n = 2\). Substitute into the formula: \(a_2 = -(-5)^{2-1} = -(-5)^1 = 5\).

Step 4: Calculate the Third Term

For the third term, set \(n = 3\). Substitute into the formula: \(a_3 = -(-5)^{3-1} = -(-5)^2 = -25\).

Step 5: Calculate the Fourth Term

For the fourth term, set \(n = 4\). Substitute into the formula: \(a_4 = -(-5)^{4-1} = -(-5)^3 = 125\).

Key Concepts

Power Functions

Power functions are an essential part of algebra and pre-calculus. They involve expressions where a variable is raised to a constant power. In simple terms, if you have a function of the form \(f(x) = ax^n\), you have a power function. Here, \(a\) is a constant, \(x\) is the variable, and \(n\) is a non-negative integer.In the original exercise, we see a sequence formula, \(a_n = -(-5)^{n-1}\), including a power function, where \(-5\) is raised to the power of \((n-1)\). This demonstrates how power functions can be part of more complex algebraic expressions beyond simple polynomials. Understanding power functions helps:

• Recognize different types of growth – linear, quadratic, or cubic.
• Solve algebraic equations involving variables raised to a power.
• Analyze behaviors of sequences and series involving powers.

Whenever a power function is involved, examine the base (like \(-5\)) and the exponent (such as \(n-1\)), as both influence the function's growth and behavior.

Exponents

Exponents are a shorthand way to represent repeated multiplication of the same factor. For instance, \(5^3\) means multiplying 5 by itself three times: \(5 \times 5 \times 5\). Exponents play a critical role in simplifying expressions and solving equations. They show up everywhere in mathematics and are especially important in sequences and series.In our sequence formula, \(-(-5)^{n-1}\), the exponent \(n-1\) dictates how many times the base \(-5\) is multiplied by itself. One intriguing part of this formula is the negative sign outside of the power expression, indicating that the result is always multiplied by \(-1\). Thus, every calculation respects the rules of exponents:

• \(a^0 = 1\) for any nonzero \(a\).
• \(a^1 = a\).
• \(a^m \times a^n = a^{m+n}\).
• \((a^m)^n = a^{m \cdot n}\).

Understanding exponents will help you evaluate sequences efficiently and recognize patterns or oscillations, like the alternating positive and negative terms in the exercise sequence.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that define a specific value or set of values. Variables serve as placeholders that can take on various values. In the expression \(a_n = -(-5)^{n-1}\), \(a_n\) is a function depending on the variable \(n\).Breaking this expression apart, we see:

• The base \(-5\) as a constant factor influenced by \(n-1\), the exponent.
• The negative sign leading the expression, flipping the sign of the resulting power.

Multiple components collaborate to determine the term's value in this sequence. Evaluating each term involves substituting \(n\) with integers \(1, 2, 3, 4,\) and so forth. This helps build an understanding of the sequence's progression and patterns.To simplify and internally resolve algebraic expressions, practice is key. By frequently working with variables, exponents, and constants, you'll develop a sharper intuition for problem-solving and expression manipulation.