Question

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \ldots\)

Short answer

The expression for the \(n\) th term of the sequence is \(a_n = (-1)^{n+1}\left(\frac{1}{n^2}\right)\).

Step-by-step solution

Identify Pattern

Look at the given sequence: \(1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \ldots\). It consists of alternating positive and negative numbers. The numbers themselves follow the pattern of reciprocals of square numbers. So extract the pattern that the nth term \(a_n\) follows.

Determine the Reciprocal Square Pattern

The series of numbers \(1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}\) suggests a pattern related to the reciprocal of a square number. For instance, the first term \(1\) can be written as \(\frac{1}{1^2}\), the second term \(\frac{1}{4}\) can be written as \(\frac{1}{2^2}\), the third term \(\frac{1}{9}\) can be written as \(\frac{1}{3^2}\), and so on. So, one part of the pattern for a term in the sequence appears to be \(\frac{1}{n^2}\) where \(n\) represents the position of the term in the sequence.

Determine the Alternating Sign Pattern

Note that the signs in the sequence alternate between positive and negative. To express this mathematically, one could use \((-1)^{n+1}\) or \((-1)^{n}\), depending on whether the first term is positive or negative. Seeing as the first term in this sequence is positive, the pattern for a term in the sequence would be expressed with \((-1)^{n+1}\). Thus, the general formula for a term in this sequence combines both patterns.

Write the General Formula

Combining the pattern of reciprocal squares and alternating signs, get a full expression for the nth term of the sequence. The formula for the nth term \(a_n = (-1)^{n+1}\left(\frac{1}{n^2}\right)\). This formula will give the nth term of the sequence for any positive integer value of \(n\).