Problem 136 - 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 5 terms of the sequence are: \(- \frac{x^{2}}{2}\), \(\frac{x^{3}}{3}\), \(- \frac{x^{4}}{4}\), \(\frac{x^{5}}{5}\), \(- \frac{x^{6}}{6}\)

Step 1: Identify the sequence formula

The given sequence has the formula \(a_{n}=\frac{(-1)^{n} x^{n+1}}{n+1}\). Next, terms are calculated by substituting integers starting from 1 into the formula in place of n.

Step 2: Calculate the first term

Substitute n = 1 into the formula to get the first term. \(a_{1}=\frac{(-1)^{1} x^{1+1}}{1+1}\) = \(- \frac{x^{2}}{2}\)

Step 3: Calculate the second term

Substitute n = 2 into the formula to get the second term. \(a_{2}=\frac{(-1)^{2} x^{2+1}}{2+1}\) = \(\frac{x^{3}}{3}\)

Step 4: Calculate the third term

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

Step 5: Calculate the fourth term

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

Step 6: Calculate the fifth term

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

Key Concepts

Algebraic Sequences

An algebraic sequence is a list of numbers formed by following a rule that is based on an algebraic expression. Each number in the sequence is known as a term. For example, the sequence given by the formula \(a_{n}=\frac{{(-1)^{n} x^{n+1}}}{{n+1}}\) is algebraic because it relies on an algebraic formula involving the variable 'n' to determine each term.

In this type of sequence, the nth term is calculated by plugging in the position number, 'n,' into the formula. For instance, to find the first five terms, you would substitute 'n' with 1, 2, 3, 4, and 5 respectively. This provides a systematic approach to find any term in the sequence, not just those in the beginning.

• The first term is found when \(n=1\)
• The second term uses \(n=2\)
• Similarly, subsequent terms follow the pattern by increasing 'n' by one each time.

Sequence Formulas

A sequence formula, also known as a general formula, serves as a recipe for computing the terms of a sequence. In our exercise, the sequence is governed by the formula \(a_{n}=\frac{{(-1)^{n} x^{n+1}}}{{n+1}}\), where 'n' represents the position of the term within the sequence.

The sequence formula condenses the pattern of the sequence into a compact algebraic expression, making it possible to find any term without listing all the previous ones. For the sequence at hand, the signs alternate due to the \( (-1)^{n} \) factor, and the power of 'x' increases by one as we move to the next term. Importantly, the denominator also changes in tandem, being exactly one unit more than the term's position 'n'.

Advantages of sequence formulas include:

• Quickly determining any term's value.
• Analyzing properties of the sequence, such as convergence or divergence.
• Facilitating the study of sequences in more complex mathematical contexts.

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of statements. While it's not directly used to calculate terms of a sequence, it is often employed to prove properties about them. The principle of induction is based on two steps:

• Base Case: Verify the statement is true for the first number in the sequence, typically when \(n=1\).
• Inductive Step: Assume the statement holds for some arbitrary positive integer \(n=k\), then show it must also be true for \(n=k+1\).

If both steps are satisfied, we can conclude the statement is true for all natural numbers 'n'. This method is analogous to a row of dominoes falling — if you knock over the first one (base case), and show that a domino falling will always knock over the next one (inductive step), you can expect all dominos to fall in sequence.