Problem 140 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The first five terms of the sequence are: \(a_{1} = -\frac{1}{2}x^2\), \(a_{2} = \frac{1}{6}x^3\), \(a_{3} = -\frac{1}{24}x^4\), \(a_{4} = \frac{1}{120}x^5\), \(a_{5} = -\frac{1}{720}x^6\)

Step 1: Substituting n=1 into the formula

Substitute \(n = 1\) into the sequence formula \(a_{n}=\frac{(-1)^{n} x^{n+1}}{(n+1) !}\) to obtain the first term \(a_{1} = \frac{-1*x^2}{2} = -\frac{1}{2}x^2\)

Step 2: Substituting n=2 into the formula

Substitute \(n = 2\) into the formula to obtain the second term \(a_{2} = \frac{(-1)^{2} x^{3}}{3!} = \frac{1}{6}x^3\)

Step 3: Substituting n=3 into the formula

Substitute \(n = 3\) into the formula to get the third term \(a_{3} = \frac{(-1)^{3} x^{4}}{4!} = -\frac{1}{24}x^4\)

Step 4: Substituting n=4 into the formula

Substitute \(n = 4\) into the formula to get the fourth term \(a_{4} = \frac{(-1)^{4} x^{5}}{5!} = \frac{1}{120}x^5\)

Step 5: Substituting n=5 into the formula

Substitute \(n = 5\) into the formula to get the fifth term \(a_{5} = \frac{(-1)^{5} x^{6}}{6!} = -\frac{1}{720}x^6\)

Key Concepts

Mathematical sequence

A mathematical sequence is a collection of numbers arranged in a specific order, where each number is typically defined by a specific rule or formula. These numbers are known as terms of the sequence.

Sequences can be finite, with a limited number of terms, or infinite, where the sequence continues indefinitely. The formula to calculate the terms of a sequence is integral to understanding its behavior and how it progresses. In the provided exercise, the sequence is defined by a formula involving both exponentiation and factorial notation, which leads to alternating positive and negative values of the terms. By identifying the pattern and applying the formula correctly, students can find any term in the sequence.

Substitution method

The substitution method is a fundamental process in mathematics used to find the value of an expression by replacing its variables with numerical values. The step-by-step solutions provided demonstrate the substitution method in action, where the variable 'n' is replaced successively with the numbers 1 through 5.

This approach is systematic and allows students to see the direct effect of changing the variable 'n' on the sequence's terms. The substitution method makes it easier to grasp the relationship between the variable and the resultant value, a skill that's valuable not just in sequences but in various mathematical contexts.

Factorial notation

Factorial notation is a mathematical notation that represents the product of all positive integers up to a specified number. It is denoted by an exclamation point, so 'n!' means 'n factorial.' For instance, 5! is equal to 5 × 4 × 3 × 2 × 1, which is 120.

In the exercise, the factorial appears in the denominator and plays a crucial role in determining the value of the sequence's terms. As 'n' increases, the factorial grows very quickly, which significantly affects the sequence, often causing the terms to get smaller. Understanding how factorial notation works is essential to solving problems involving sequences and many other mathematical topics where factorials are applied.