Question

Write the first five terms of the sequence. \(a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)\)

Short answer

The first five terms of the sequence are 2, -1, 2/3, -1/2, 2/5.

Step-by-step solution

Calculate first term

Use the formula to calculate the first term of the sequence by substituting \(n=1\) into the given formula: \(a_1=(-1)^{1+1}\times\left(\frac{2}{1}\right)\).\nThis simplifies to \(a_1=2\)

Calculate second term

Use the formula to calculate the second term by substituting \(n=2\) into the given formula: \(a_2=(-1)^{2+1}\times\left(\frac{2}{2}\right)\).\nThis simplifies to \(a_2=-1\)

Calculate third term

Use the formula to calculate the third term by substituting \(n=3\) into the given formula: \(a_3=(-1)^{3+1}\times\left(\frac{2}{3}\right)\).\nThis simplifies to \(a_3=\frac{2}{3}\)

Calculate fourth term

Use the formula to calculate the fourth term by substituting \(n=4\) into the given formula: \(a_4=(-1)^{4+1}\times\left(\frac{2}{4}\right)\).\nThis simplifies to \(a_4=-\frac{1}{2}\)

Calculate fifth term

Use the formula to calculate the fifth term by substituting \(n=5\) into the given formula: \(a_5=(-1)^{5+1}\times\left(\frac{2}{5}\right)\).\nThis simplifies to \(a_5=\frac{2}{5}\)

Key concepts

Sequence Formula

Mathematical sequences are like treasure maps that lead us to each number or point in an organized pattern. The sequence formula is like the key to this treasure, giving us a specific rule or equation to find any term in the sequence. For example, in the given exercise, the sequence formula is \(a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)\).

To use this formula, you replace \(n\) with the position number of the term you're after. Imagine you're playing hopscotch, and the number \(n\) tells you which square to hop to; the formula then tells you what you'll find there. This is how we calculated the first five terms, stepping from one to five and applying the formula to find out what lay in each 'square' of the sequence. Understanding how to apply the sequence formula is crucial for delving into more complex mathematics.

Arithmetic Sequences

An arithmetic sequence is a lineup of numbers where each number is found by adding a constant amount, known as the common difference, to the previous number. It’s like a staircase where each step up or down is exactly the same height. In the exercise given, the sequence does not add a fixed amount each time; instead, it alternates between positive and negative fractions and so, it’s not an arithmetic sequence.

However, understanding arithmetic sequences is still essential, as they are everywhere! Think of the daily routine where the alarm clock rings at the same time, or the pattern of a marching band stepping forward in unison. Grasping arithmetic sequences helps in solving real-world problems involving repeated and regular patterns.

Convergence of Sequences

In the world of sequences, convergence is like a story with a particular ending; the numbers in the sequence get closer and closer to a specific value as they continue. Think of it like a puppy chasing a scheduled food drop—the gap closes over time, representing the numbers in the sequence. Not all sequences converge; some are wild and keep bouncing around without settling down.

The sequence in our exercise behaves like the puppy with no intention to settle down immediately, as the fractions continue to oscillate between positive and negative values. Understanding the convergence of sequences is vital because it enables us to predict behavior in mathematics and natural phenomena, from the growth of populations to the flow of traffic, giving us a picture of stability—or a lack of it—in the pattern.