To predict the general term, or the [tex]\( n \)[/tex]-th term, [tex]\( a_n \)[/tex], of the given sequence:
[tex]\[
\frac{7}{19}, \frac{8}{20}, \frac{9}{21}, \frac{10}{22}, \frac{11}{23}, \ldots
\][/tex]
we need to identify the patterns in the numerators and the denominators of the fractions.
### Step 1: Analyze the Numerator
The numerators of the sequence are:
[tex]\[ 7, 8, 9, 10, 11, \ldots \][/tex]
This is an arithmetic sequence where the first term [tex]\( a = 7 \)[/tex] and the common difference [tex]\( d = 1 \)[/tex].
The general term for the numerator of an arithmetic sequence is given by:
[tex]\[ a_n = a + (n - 1)d \][/tex]
Substituting [tex]\( a = 7 \)[/tex] and [tex]\( d = 1 \)[/tex], we get:
[tex]\[
\text{numerator} = 7 + (n - 1) \cdot 1 = 7 + n - 1 = n + 6
\][/tex]
### Step 2: Analyze the Denominator
The denominators of the sequence are:
[tex]\[ 19, 20, 21, 22, 23, \ldots \][/tex]
This is also an arithmetic sequence where the first term [tex]\( b = 19 \)[/tex] and the common difference [tex]\( d = 1 \)[/tex].
The general term for the denominator of an arithmetic sequence is given by:
[tex]\[ b_n = b + (n - 1)d \][/tex]
Substituting [tex]\( b = 19 \)[/tex] and [tex]\( d = 1 \)[/tex], we get:
[tex]\[
\text{denominator} = 19 + (n - 1) \cdot 1 = 19 + n - 1 = n + 18
\][/tex]
### Step 3: Formulate the General Term
The general [tex]\( n \)[/tex]-th term of the given sequence is the ratio of the general terms of the numerator and the denominator we just found:
[tex]\[
a_n = \frac{n + 6}{n + 18}
\][/tex]
Therefore, the general term, or [tex]\( n \)[/tex]-th term, of the sequence is:
[tex]\[
a_n = \frac{n + 6}{n + 18}
\][/tex]
