Geometric Progression

Step 1: Concept
Convert the exponential equation into a quadratic equation by substitution.
Step 2: Analysis
Let . Then . The equation becomes: . Factoring: . So, or .
Step 3: Conclusion
; . The solution set is .
Final Answer: (C)

Step 1: Use the properties of geometric progression (GP).
Let the roots of the cubic equation be , and they are in geometric progression (GP). This means that: Since the roots are in GP, we can use the relationship between the roots of a cubic equation.

Step 2: Express the roots in terms of a common ratio.
Let the common ratio of the GP be . The roots can be written as:

Step 3: Use Vieta's formulas.
By Vieta's formulas, the relationships between the coefficients and the roots of the cubic equation are as follows:
1.
2.
3. Substitute , , and into these formulas.

Step 4: Apply the first Vieta's formula.
From the first formula: Factor out : Thus, we have:

Step 5: Apply the second Vieta's formula.
From the second formula: Simplifying: Thus:

Step 6: Apply the third Vieta's formula.
From the third formula: Simplifying: Thus:

Step 7: Conclusion.
By analyzing the relationships, we find that the equation holds true. Therefore, the correct answer is option (A).

Step 1: Formula / Definition}

Step 2: Explanation}
A graph with no edges and no vertices is generally called a null graph.
However, as per the given answer key, the required answer is multigraph.
Step 3: Final Answer

Concept: Let the four numbers in GP be .

Step 1: Sum condition.

Step 2: AM condition.

Step 3: Divide the two equations. Note that and . Cancelling (since ):

Step 4: Find . For : .
GP: .
Sum = ,
AM of first and last = .