Question

find the cardinality of the following set.
b = {1, 3, 5, ..., 25}
n(b) =

Explanation:

Step1: Identify the sequence type

The set \( B = \{1, 3, 5, \dots, 25\} \) is an arithmetic sequence where the first term \( a_1 = 1 \), the common difference \( d = 2 \), and the last term \( a_n = 25 \).

Step2: Use the arithmetic sequence formula

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \). Substitute \( a_n = 25 \), \( a_1 = 1 \), and \( d = 2 \) into the formula:

$$ 25 = 1 + (n - 1) \times 2 $$

Step3: Solve for \( n \)

First, subtract 1 from both sides:

$$ 24 = (n - 1) \times 2 $$

Then, divide both sides by 2:

$$ 12 = n - 1 $$

Finally, add 1 to both sides:

$$ n = 13 $$

Answer:

13