Question

find the cardinal number for the given set.
a = {7, 10, 13, ..., 43}

the cardinal number is
(type a whole number.)

Explanation:

Step1: Identify the sequence type

The set \( A = \{7, 10, 13, \dots, 43\} \) is an arithmetic sequence where the first term \( a_1 = 7 \), common difference \( d = 10 - 7 = 3 \), and the last term \( a_n = 43 \).

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 \( a_n = 43 \), \( a_1 = 7 \), and \( d = 3 \) into the formula:

$$ 43 = 7 + (n - 1) \times 3 $$

Step3: Solve for \( n \)

First, subtract 7 from both sides:

$$ 43 - 7 = (n - 1) \times 3 $$

$$ 36 = (n - 1) \times 3 $$

Then, divide both sides by 3:

$$ 12 = n - 1 $$

Finally, add 1 to both sides:

$$ n = 13 $$

Answer:

13