To determine the general term [tex]\( a_n \)[/tex] of the sequence [tex]\(\frac{12}{14}, \frac{13}{15}, \frac{14}{16}, \frac{15}{17}, \frac{16}{18}, \ldots \)[/tex], let's analyze the pattern of the numerators and denominators separately.
1. Numerator Pattern:
- The numerators in the sequence are: [tex]\( 12, 13, 14, 15, 16, \ldots \)[/tex]
- We can see that the numerators start at 12 and increase by 1 with each term.
- The [tex]\( n \)[/tex]-th numerator can be expressed as:
[tex]\[
\text{numerator} = 12 + (n - 1) = 11 + n
\][/tex]
2. Denominator Pattern:
- The denominators in the sequence are: [tex]\( 14, 15, 16, 17, 18, \ldots \)[/tex]
- We can see that the denominators start at 14 and also increase by 1 with each term.
- The [tex]\( n \)[/tex]-th denominator can be expressed as:
[tex]\[
\text{denominator} = 14 + (n - 1) = 13 + n
\][/tex]
Now, we can combine these patterns to express the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of the sequence, which is the ratio of the [tex]\( n \)[/tex]-th numerator to the [tex]\( n \)[/tex]-th denominator.
Thus, the general term [tex]\( a_n \)[/tex] of the sequence is:
[tex]\[
a_n = \frac{11+n}{13+n}
\][/tex]
So, the general term [tex]\( a_n \)[/tex] is:
[tex]\[
a_n = \frac{11 + n}{13 + n}
\][/tex]
