91Ó°ÊÓ

In a geometric sequence, the common ratio is the fixed number that each term is multiplied by to get the next term. This ratio is the backbone of the sequence, ensuring each term properly relates to its predecessor. For instance, with the sequence \(a, ar, ar^2, \ldots\), the common ratio is \(r\). This consistent multiplication makes the sequence geometric.
For the specific terms \(a, \sqrt{ab}, b\), we identify the common ratio by equating the ratio of successive terms and simplifying it. Using the first two terms, \(\frac{\sqrt{ab}}{a}\), and the second and third terms, \(\frac{b}{\sqrt{ab}}\), we equate \(\frac{\sqrt{ab}}{a} = \frac{b}{\sqrt{ab}}\). By simplifying, we find that both fractions lead to the same expression. Thus, they demonstrate a consistent common ratio, affirming a valid geometric sequence exists.

Successive terms in a geometric sequence are closely related by the common ratio. They come one after another, each one derived directly from the previous term by multiplication with the ratio. The property of successive terms ensures the term-to-term structure of the sequence.

• In our example, successive terms are \(a, \sqrt{ab}, b\).
• To ensure these form a geometric sequence, we express both successive ratios: \(\frac{\sqrt{ab}}{a}\) and \(\frac{b}{\sqrt{ab}}\).
  

Both should simplify to give the same result reflecting a constant ratio, thereby confirming they are true successive terms in a geometric sequence.

Simplifying the expressions \(\frac{\sqrt{ab}}{a}\) and \(\frac{b}{\sqrt{ab}}\) helps verify the terms belong to a geometric sequence. Simplification involves reducing a complex fraction to its simplest form, revealing underlying patterns and relationships.
For our terms:

• The left side \( \frac{\sqrt{ab}}{a} \) simplifies as \(\frac{ab^{1/2}}{a} = \frac{b^{1/2}}{a^{1/2}}\).
• The right side \(\frac{b}{\sqrt{ab}}\) simplifies similarly to yield \(\frac{b^{1/2}}{a^{1/2}}\).

Both sides simplify to the same expression \(\frac{b^{1/2}}{a^{1/2}}\), proving the terms \(a, \sqrt{ab}, b\) indeed maintain a constant ratio. Simplification is essential for establishing and verifying the sequence.

A mathematical proof is an argument that establishes the truth of a mathematical statement. In sequence problems, proofs usually involve demonstrating properties using logical step-by-step progressions. The goal is to show that specific criteria are met.
To prove \(a, \sqrt{ab}, b\) form a geometric sequence, we followed these steps:

• Defined the terms and expressed their ratios.
• Simplified expressions to show equality of the resulting common ratio.
• Concluded the consistent ratio implied the terms belong to a geometric sequence.

Clearly illustrating each step and reaching a logical conclusion confirms the sequence's properties. Proofs ensure the validity of mathematical findings.