Let \(a, b, b, c,\) and \(d\) be distinct positive integers. What is the 4th term of the geometric sequence below? $$b c d, a b c^{2} d, a^{2} b c^{3} d, \cdots$$ F. \(a^{3} b c^{4} d\) G. \(a^{3} b^{2} c^{3} d\) H. \(a^{3} b^{2} c^{4} d^{2}\) J. \(a^{4} b c^{6} d\) K. \(a^{4} b c^{9} d\)

In a geometric sequence, the common ratio is one of the fundamental aspects that defines the sequence's character. It's the factor by which we multiply each term to get the next one. Understanding the common ratio is crucial because it helps us to determine the progression of the sequence and predict future terms.

How do we find this common ratio? It's simple: we pick two consecutive terms in the sequence, and divide the latter by the former. For instance, consider the sequence \( 2, 6, 18, 54, ... \). Dividing the second term (6) by the first (2) or the third term (18) by the second (6) consistently gives us 3 as the common ratio. This means that to move from one term to the next, we multiply by 3.

When we calculate sequence term calculation, we're essentially trying to find the value of a particular term in a sequence. With geometric sequences, each term is derived by multiplying the previous term by the common ratio. This systematic approach allows us to calculate any term in the sequence, regardless of its position, as long as we know the initial term and the common ratio.

For example, let's say we wish to find the 5th term of the geometric sequence starting with 2 and a common ratio of 3. We can simply multiply the first term by the common ratio raised to the power of the term's position minus one: \( 2 \cdot 3^{4} \). The power indicates how many times we've applied the common ratio starting from the initial term. This is a powerful technique that enables us to leap to any position in the sequence without laboriously multiplying term by term.

Lastly, let's talk about algebraic expressions. They are combinations of numbers, variables (like \( x \), \( y \), etc.), and operators (like \( + \), \( - \), \( \cdot \) and \( ÷ \) ) that show the relationship between quantities. They can take many forms, from simple linear expressions to more complex polynomial expressions. In the context of geometric sequences, we deal with algebraic expressions that often involve terms raised to various powers, which represent the progression of the sequence.

Understanding how to manipulate these expressions is crucial. Simplifying or expanding them allows us to identify patterns, solve equations, and generally work more effectively with mathematical relationships. For instance, simplifying the expression \( a^2(bc) \cdot a^3(b^2c^3) \) involves combining like terms and using exponent rules to obtain the final simplified form.