The sequence shown in the image is:

$2^{\frac{3}{2}}, \quad 2^{\frac{5}{4}}, \quad 2^{\frac{7}{8}}, \quad \dots$

Implicit Function (Recursive Relation)

The sequence can be described implicitly using a recursive relation:

$a_{n+1} = 2^{\frac{3}{2^n}}, \quad \text{where } a_1 = 2^{\frac{3}{2}} \text{ and } n \geq 1$

Here, each term is a power of 2 with an exponent that follows a recursive pattern.

Explicit Function

The explicit function for the $n$-th term of the sequence can be written as:

$a_n = 2^{\frac{2n+1}{2^n}}, \quad \text{where } n \geq 1$

Explanation:

• The first term is $2^{\frac{3}{2}}$.
• The second term is $2^{\frac{5}{4}}$.
• The third term is $2^{\frac{7}{8}}$.
• Continuing this pattern, the exponent in the $n$-th term is $\frac{2n+1}{2^n}$.

Would you like a detailed step-by-step explanation of how these functions were derived, or do you have any specific questions about them?

Related Questions:

1. How do you determine the pattern in the exponents of the sequence?
2. What is the general approach to finding explicit formulas for sequences?
3. Can this sequence be related to a geometric series or another known series?
4. What is the sum of the sequence if it's finite?
5. How does this sequence behave as $n$ approaches infinity?

Tip: When dealing with sequences, always check if there's a pattern in the differences or ratios between consecutive terms—it often provides clues for the explicit function.