Get expert insights and reliable answers to your questions on IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.
Sagot :
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.
[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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.