In a geometric sequence, each term is found using the formula:

$a_n = a_1 \cdot r^{n-1}$

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the position of the term in the sequence.

We are given:

• The 3rd term $a_3 = 18$,
• The 7th term $a_7 = 1458$.

Let's set up the equations for the 3rd and 7th terms.

For the 3rd term:

$a_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2 = 18$

For the 7th term:

$a_7 = a_1 \cdot r^{7-1} = a_1 \cdot r^6 = 1458$

So, we have the system of two equations:

1. $a_1 \cdot r^2 = 18$
2. $a_1 \cdot r^6 = 1458$

Step 1: Solve for $a_1$ in terms of $r$

From equation (1), we can express $a_1$ as:

$a_1 = \frac{18}{r^2}$

Step 2: Substitute into equation (2)

Substitute this expression for $a_1$ into the second equation:

$\frac{18}{r^2} \cdot r^6 = 1458$

Simplify the equation:

$18 \cdot r^4 = 1458$

Step 3: Solve for $r^4$

Now divide both sides by 18:

$r^4 = \frac{1458}{18} = 81$

Step 4: Solve for $r$

Take the fourth root of both sides:

$r = \sqrt[4]{81} = 3$

Step 5: Find $a_1$

Now that we know $r = 3$, substitute this back into the equation for $a_1$:

$a_1 = \frac{18}{r^2} = \frac{18}{3^2} = \frac{18}{9} = 2$

So, the first term is $\boxed{2}$.

Would you like any further details on this solution? Or do you have another question?

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Here are some related questions you might find interesting:

1. What is the common ratio $r$ for the geometric sequence?
2. How can we use the formula for the $n$-th term of a geometric sequence to find specific terms?
3. How does the common ratio affect the growth or decay of the sequence?
4. How can the geometric mean be calculated from two terms of a geometric sequence?
5. What is the general form of the sum of a geometric series?

Tip: When working with geometric sequences, it's often helpful to use logarithms if you need to solve for exponents.