Answer :
To identify the expression for the [tex]\(n\)[/tex]-th term in the given pattern, let's first analyze the sequence:
- The first term is [tex]\(1 = \frac{1}{1^2}\)[/tex].
- The second term is [tex]\(\frac{1}{4} = \frac{1}{2^2}\)[/tex].
- The third term is [tex]\(\frac{1}{9} = \frac{1}{3^2}\)[/tex].
- The fourth term is [tex]\(\frac{1}{16} = \frac{1}{4^2}\)[/tex].
- The fifth term is [tex]\(\frac{1}{25} = \frac{1}{5^2}\)[/tex].
We can observe that each term of the sequence can be written as a fraction where the numerator is always [tex]\(1\)[/tex] and the denominator is the square of a natural number. More specifically, the denominators are [tex]\(1^2, 2^2, 3^2, 4^2, 5^2,\ldots\)[/tex].
Hence, for the [tex]\(n\)[/tex]-th term in the sequence, the denominator is [tex]\(n^2\)[/tex].
Therefore, the general expression for the [tex]\(n\)[/tex]-th term of the sequence is:
[tex]\[ \frac{1}{n^2} \][/tex]
Thus, the expression for the [tex]\(n\)[/tex]-th term in this pattern is [tex]\(\boxed{\frac{1}{n^2}}\)[/tex].
- The first term is [tex]\(1 = \frac{1}{1^2}\)[/tex].
- The second term is [tex]\(\frac{1}{4} = \frac{1}{2^2}\)[/tex].
- The third term is [tex]\(\frac{1}{9} = \frac{1}{3^2}\)[/tex].
- The fourth term is [tex]\(\frac{1}{16} = \frac{1}{4^2}\)[/tex].
- The fifth term is [tex]\(\frac{1}{25} = \frac{1}{5^2}\)[/tex].
We can observe that each term of the sequence can be written as a fraction where the numerator is always [tex]\(1\)[/tex] and the denominator is the square of a natural number. More specifically, the denominators are [tex]\(1^2, 2^2, 3^2, 4^2, 5^2,\ldots\)[/tex].
Hence, for the [tex]\(n\)[/tex]-th term in the sequence, the denominator is [tex]\(n^2\)[/tex].
Therefore, the general expression for the [tex]\(n\)[/tex]-th term of the sequence is:
[tex]\[ \frac{1}{n^2} \][/tex]
Thus, the expression for the [tex]\(n\)[/tex]-th term in this pattern is [tex]\(\boxed{\frac{1}{n^2}}\)[/tex].
