Find $a_{1}$ and $r$ for a geometric sequence with the values given below. \[ a_{n}=24, n=4, S_{n}=45 \] $a_{1}=\square$ (Type an integer or a simplified fraction.)

Step 1 ：Given that the fourth term of the geometric sequence is 24 and the sum of the first four terms is 45, we can write these facts as equations using the formulas for the nth term and the sum of the first n terms of a geometric sequence.

Step 2 ：The nth term of a geometric sequence is given by \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. So, we have \(a_4 = a_1 \cdot r^{(4-1)} = 24\).

Step 3 ：The sum of the first n terms of a geometric sequence is given by \(S_n = a_1 \cdot \frac{1 - r^n}{1 - r}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. So, we have \(S_4 = a_1 \cdot \frac{1 - r^4}{1 - r} = 45\).

Step 4 ：We can solve these two equations simultaneously to find the values of \(a_1\) and \(r\).

Step 5 ：From the first equation, we can express \(a_1\) in terms of \(r\) as \(a_1 = \frac{24}{r^3}\).

Step 6 ：Substitute \(a_1 = \frac{24}{r^3}\) into the second equation, we get \(45 = \frac{24}{r^3} \cdot \frac{1 - r^4}{1 - r}\).

Step 7 ：Solving this equation for \(r\), we get \(r = 2\).

Step 8 ：Substitute \(r = 2\) into the equation \(a_1 = \frac{24}{r^3}\), we get \(a_1 = 3\).

Step 9 ：So, the first term \(a_1\) of the geometric sequence is 3 and the common ratio \(r\) is 2.