To find the \( n \)th term of the number sequence \( 2, 4, 6, 8, \ldots \):
1. Identify the sequence type:
This is an arithmetic sequence because the difference between consecutive terms is constant.
2. Determine the first term \( a \) and the common difference \( d \):
- The first term \( a \) is \( 2 \).
- The common difference \( d \) is found by subtracting the first term from the second term: \( 4 - 2 = 2 \).
3. Write the formula for the \( n \)th term of an arithmetic sequence:
The general formula for the \( n \)th term \( a_n \) of an arithmetic sequence is:
[tex]\[
a_n = a + (n - 1)d
\][/tex]
Here, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the position of the term in the sequence.
4. Substitute the known values into the formula:
- First term \( a = 2 \)
- Common difference \( d = 2 \)
[tex]\[
a_n = 2 + (n - 1) \times 2
\][/tex]
5. Simplify the expression:
[tex]\[
a_n = 2 + 2n - 2
\][/tex]
[tex]\[
a_n = 2n
\][/tex]
6. Conclusion:
The \( n \)th term of the sequence \( 2, 4, 6, 8, \ldots \) is:
[tex]\[
a_n = 2n
\][/tex]
So, if for example you wanted to find the 5th term (\( n = 5 \)) of this sequence, you would calculate:
[tex]\[
a_5 = 2 \times 5 = 10
\][/tex]
The result is [tex]\( 10 \)[/tex].
