To determine if 127 is in the given sequence, we need to analyze the pattern of the sequence. The given sequence is:
[tex]\[ 3, 8, 13, \ldots \][/tex]
We can observe that the sequence is arithmetic because the difference between consecutive terms is constant. Let's identify the terms:
- The first term ([tex]\(a_1\)[/tex]) is 3.
- The common difference ([tex]\(d\)[/tex]) can be found by subtracting the first term from the second term:
[tex]\[ d = 8 - 3 = 5 \][/tex]
The general form for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
We want to know if 127 is in the sequence. Therefore, we need to solve for [tex]\(n\)[/tex] in the equation:
[tex]\[ 127 = 3 + (n-1) \cdot 5 \][/tex]
First, isolate the term involving [tex]\(n\)[/tex]:
[tex]\[ 127 - 3 = (n-1) \cdot 5 \][/tex]
[tex]\[ 124 = (n-1) \cdot 5 \][/tex]
Next, solve for [tex]\(n-1\)[/tex] by dividing both sides of the equation by the common difference, 5:
[tex]\[ \frac{124}{5} = n - 1 \][/tex]
Performing the division:
[tex]\[ 24.8 = n - 1 \][/tex]
Then, solve for [tex]\(n\)[/tex] by adding 1 to both sides:
[tex]\[ n = 24.8 + 1 \][/tex]
[tex]\[ n = 25.8 \][/tex]
Since [tex]\(n\)[/tex] needs to be an integer (as [tex]\(n\)[/tex] represents the term index in the sequence), and 25.8 is not an integer, we conclude that 127 is not a term in the sequence.
Thus, 127 is not in the sequence.
