To determine the number of elements in the set [tex]\( B = \{2, 4, 6, \ldots, 22\} \)[/tex], we recognize that the set forms an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term.
Here's the step-by-step solution:
1. Identify the first term and the common difference:
- The first term [tex]\( a \)[/tex] is 2.
- The common difference [tex]\( d \)[/tex] is the difference between consecutive terms, which is 2 (i.e., [tex]\( 4 - 2 = 2 \)[/tex]).
2. Identify the last term:
- The last term [tex]\( l \)[/tex] in the set is 22.
3. Use the formula for the number of terms in an arithmetic sequence:
The formula to find the number of terms [tex]\( n \)[/tex] in an arithmetic sequence is:
[tex]\[
n = \frac{l - a}{d} + 1
\][/tex]
where [tex]\( l \)[/tex] is the last term, [tex]\( a \)[/tex] is the first term, and [tex]\( d \)[/tex] is the common difference.
4. Substitute the values:
- First term [tex]\( a = 2 \)[/tex]
- Last term [tex]\( l = 22 \)[/tex]
- Common difference [tex]\( d = 2 \)[/tex]
Substituting these values into the formula gives:
[tex]\[
n = \frac{22 - 2}{2} + 1
\][/tex]
5. Calculate the expression step-by-step:
- Subtract the first term from the last term:
[tex]\[
22 - 2 = 20
\][/tex]
- Divide the result by the common difference:
[tex]\[
\frac{20}{2} = 10
\][/tex]
- Add 1 to the quotient:
[tex]\[
10 + 1 = 11
\][/tex]
Therefore, the number of elements in the set [tex]\( B = \{2, 4, 6, \ldots, 22\} \)[/tex] is [tex]\( 11 \)[/tex].
