91影视

In sequences, alternating signs are used to create a pattern where the terms switch between positive and negative values. This sequence behavior is captured in the numerator of the formula, particularly by the expression \[(-1)^{n+1}\]Here:

• If \(n\) is odd (like \(n = 1, 3, 5, \) etc.), \((-1)^{n+1}\) becomes \((-1)^2 = 1\), which is positive.
• If \(n\) is even (like \(n = 2, 4, 6, \) etc.), \((-1)^{n+1}\) becomes \((-1)^3 = -1\), which is negative.

This technique of using powers of -1 is a common method in mathematics to alternate signs in sequences and series. By understanding this, you can predict the sign of any term in the sequence.

The sequence is defined by a general formula that generates its terms depending on the position \(n\). Here the formula given is:\[a_{n} = \frac{(-1)^{n+1}}{2n-1}\]The term "nth term" refers to the formula's ability to derive any term in the sequence when substituting different values of \(n\). This concept is useful when you want to calculate terms without having to sequentially compute each preceding one.Each \(n\) corresponds to one term in the sequence: \(a_1\), \(a_2\), \(a_3\), and so on. By plugging in values of \(n\), you can obtain corresponding terms of the sequence efficiently:

• \(a_1\) when \(n = 1\)
• \(a_2\) when \(n = 2\)
• \(a_3\) when \(n = 3\)
• \(a_4\) when \(n = 4\)

This is the fundamental principle behind any sequence: to produce terms methodically through the nth term formula.

The denominator in a sequence formula plays a crucial role in determining the values of the sequence's terms. In our given sequence, it is defined as:\[2n-1\]This expression helps to ensure that the terms have different magnitudes, as each term is calculated by dividing the alternating sign by this value. As \(n\) increases, the denominator also increases, affecting the term's magnitude.Here's how it works:

• When \(n = 1\), the denominator is \(2 \times 1 - 1 = 1\).
• When \(n = 2\), it becomes \(2 \times 2 - 1 = 3\).
• When \(n = 3\), it changes to \(2 \times 3 - 1 = 5\).
• Continuing this pattern, \(n = 4\) yields a denominator of \(2 \times 4 - 1 = 7\).

The denominator affects the size of each term while the sign is governed by the numerator. Understanding both aspects is important for fully grasping how each sequence term is derived.