To find the \( n \)th term of the sequence \( 6, 11, 16, 21, \ldots \), we recognize that this is an arithmetic sequence. An arithmetic sequence is defined by a first term (usually denoted as \( a \)) and a common difference (denoted as \( d \)) between consecutive terms.
1. Identify the First Term and Common Difference:
- The first term \( a \) of our sequence is 6.
- The common difference \( d \) can be calculated by subtracting the first term from the second term:
[tex]\[
d = 11 - 6 = 5
\][/tex]
2. Write the General Formula:
The formula for the \( n \)th term of an arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots \) is:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
3. Substitute the Values into the Formula:
Using the values we identified:
- \( a = 6 \)
- \( d = 5 \)
The formula for the \( n \)th term becomes:
[tex]\[
a_n = 6 + (n - 1) \cdot 5
\][/tex]
This gives us the \( n \)th term of the sequence \( 6, 11, 16, 21, \ldots \).
For example, to find the 10th term of the sequence, substitute \( n = 10 \) into the formula:
[tex]\[
a_{10} = 6 + (10 - 1) \cdot 5
\][/tex]
[tex]\[
a_{10} = 6 + 9 \cdot 5
\][/tex]
[tex]\[
a_{10} = 6 + 45
\][/tex]
[tex]\[
a_{10} = 51
\][/tex]
Therefore, the 10th term of the sequence is 51.
