Problem 137 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence. $$a_{n}=\frac{(-1)^{n} x^{2 n}}{(2 n) !}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are: \(a_{1}=-\frac{x^{2}}{2}\), \(a_{2}=\frac{x^{4}}{24}\), \(a_{3}= -\frac{x^{6}}{720}\), \(a_{4}=\frac{x^{8}}{40320}\), \(a_{5}= -\frac{x^{10}}{3628800}\)

Step 1: Substitute n=1 into the sequence

When you substitute n=1 into the sequence formula, the equation becomes \(a_{1} = \frac{(-1)^{1} x^{2\cdot1}}{(2\cdot1)!} \) which simplifies to \(a_{1}=-\frac{x^{2}}{2}\)

Step 2: Substitute n=2 into the sequence

Substitute n=2 into the sequence formula. The equation becomes \(a_{2} = \frac{(-1)^{2} x^{2\cdot2}}{(2\cdot2)!}\) which simplifies to \(a_{2}=\frac{x^{4}}{24}\)

Step 3: Substitute n=3 into the sequence

When you substitute n=3 into the sequence formula, the equation becomes \(a_{3} = \frac{(-1)^{3} x^{2\cdot3}}{(2\cdot3)!}\) which simplifies to \(a_{3}= -\frac{x^{6}}{720}\)

Step 4: Substitute n=4 into the sequence

Substitute n=4 into the sequence formula. The equation becomes \(a_{4} = \frac{(-1)^{4} x^{2\cdot4}}{(2\cdot4)!}\) which simplifies to \(a_{4}=\frac{x^{8}}{40320}\)

Step 5: Substitute n=5 into the sequence

When you substitute n=5 into the sequence formula, the equation becomes \(a_{5} = \frac{(-1)^{5} x^{2\cdot5}}{(2\cdot5)!}\) which simplifies to \(a_{5}= -\frac{x^{10}}{3628800}\)

Key Concepts

Factorial

A factorial, denoted by an exclamation mark (!) following a number, is the product of all positive integers up to that number. It's a way of multiplying a sequence of descending natural numbers. Factorials are particularly important in permutations, combinations, and various mathematical sequences. For example:

• 4! = 4 × 3 × 2 × 1 = 24
• 5! = 5 × 4 × 3 × 2 × 1 = 120

In this sequence formula, the factorial appears in the denominator as \( (2n)! \). As \( n \) increases, the factorial grows very fast, which greatly affects the size of the term. This factorial is crucial in calculus and series expansions, as it helps to rapidly reduce the size of terms in an infinite series.

Alternating Series

An alternating series is a sequence where the signs alternate between positive and negative. In our sequence, we see this structure from the \((-1)^n\) term. Here's how it affects the sequence:

• When \( n \) is odd, \((-1)^n\) results in a negative sign.
• When \( n \) is even, \((-1)^n\) results in a positive sign.

This feature causes the sequence to alternate in sign, displaying a back-and-forth pattern. Alternating series are common in mathematics because their convergence properties allow for unique mathematical applications and analysis.

Exponents

Exponents provide a way to express repeated multiplication of a number by itself. In this sequence, exponents appear in the numerator of the sequence term as \( x^{2n} \). More specifically:

• The base \( x \) is repeatedly multiplied by itself \( 2n \) times.
• For the first term, \( x^2\) represents \( x \) times \( x \).
• In the second term, \( x^4 \) means \( x \times x \times x \times x \).

Exponents in mathematical sequences, especially when growing with variables like \( n \), can drastically increase the size of each term. This growth underscores the importance of understanding exponential functions, as they can impact the behavior and convergence of a sequence.

Term Simplification

Term simplification is the process of making an expression or equation simpler and more readable. We use simplification throughout mathematics to make calculations easier and more intuitive. The given sequence uses simplification in each of its terms to reveal its core structure:

• Initially, replace \( n \) with specific values to calculate each term, such as \( n = 1, 2, 3, \) etc.
• Calculate each component separately: \((-1)^n\), \(x^{2n}\), and \((2n)!\).
• Combine and simplify the calculation step by step to reduce to the simplest fractional form.

By simplifying terms, you gain clarity and can see the underlying relationships more easily. It helps reveal the alternating nature and the growth patterns within the sequence.