Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}

Verified step by step guidance

1. First, observe the pattern of the sequence: the terms alternate in sign and increase in absolute value by 1 each time. The sequence is {1, -2, 3, -4, 5, -6, ...}. Note that the sign alternates and the magnitude is simply the term's position in the sequence.
2. To capture the alternating sign, use the factor \((-1)^{n+1}\) or \((-1)^{n}\), since these expressions alternate between +1 and -1 as \(n\) increases. For example, \((-1)^{n+1}\) is positive when \(n\) is odd and negative when \(n\) is even.
3. The magnitude of each term is just \(n\), the position in the sequence. So one explicit formula can be written as \(a_n = (-1)^{n+1} \times n\).
4. For a second explicit formula, consider using the floor or ceiling functions or an expression involving powers of \(-1\) in a different way. For example, you can write \(a_n = (-1)^{n+1} \cdot n\) or \(a_n = (-1)^{n+1} \cdot n\) (which is the same), so try expressing the sign as \$1 - 2(n \bmod 2)\( or \)(-1)^n\( with a shift in \)n$ to get the sign pattern.
5. Another way is to write the sign as \((-1)^{n+1}\) and the magnitude as \(n\), or alternatively, use \(a_n = (-1)^{n+1} \cdot n\) and \(a_n = (-1)^{n+1} \cdot n\) but with \(n\) replaced by \(n\) or \(n\) shifted by 1, depending on the indexing. The key is to express the alternating sign and the magnitude explicitly.

Key Concepts

Sequences and Explicit Formulas

A sequence is an ordered list of numbers following a specific pattern. An explicit formula defines the nth term directly as a function of n, allowing calculation of any term without knowing previous terms.

Alternating Sign Patterns

Alternating sign sequences switch between positive and negative values, often modeled using powers of -1, such as (-1)^n or (-1)^{n+1}, to represent the sign change at each term.

Piecewise and General Term Construction

Constructing explicit formulas may involve combining absolute values, powers, or piecewise definitions to capture complex patterns, such as alternating signs combined with increasing magnitudes.