Question

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

Accepted Answer

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. 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. 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}$$ 	imes n. $$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. 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.

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

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Key Concepts

Sequences and Explicit Formulas

Alternating Sign Patterns

Piecewise and General Term Construction