Question

find the cardinal number for the given set. a = {1, 5, 9,..., 53} the cardinal number is . (type a whole number.)

Explanation:

Step1: Identify the sequence type

The set forms an arithmetic - sequence with first term $a_1 = 1$ and common difference $d=4$.

Step2: Use the formula for the $n$th term of an arithmetic sequence

The formula for the $n$th term of an arithmetic sequence is $a_n=a_1+(n - 1)d$. We know $a_n = 53$, $a_1 = 1$ and $d = 4$. Substitute these values into the formula: $53=1+(n - 1)\times4$.

Step3: Solve the equation for $n$

First, simplify the equation:
\[

$$\begin{align*} 53&=1 + 4n-4\\ 53&=4n-3\\ 4n&=53 + 3\\ 4n&=56\\ n&=14 \end{align*}$$

\]
The cardinal number of a set is the number of elements in the set. Here, $n$ represents the number of elements.

Answer:

14