To solve this problem, we are tasked with finding the explicit formula equivalent to the given recurrence relation:

$a_1 = 7, \quad a_n = a_{n-1} + 3.2$

Step 1: Understanding the formula

The recurrence relation defines an arithmetic sequence, where the first term $a_1 = 7$, and each subsequent term increases by the common difference $d = 3.2$.

For an arithmetic sequence, the general explicit formula is given by:

$a_n = a_1 + (n-1)d$

Step 2: Substituting values

Here, $a_1 = 7$ and $d = 3.2$. Substituting these into the explicit formula:

$a_n = 7 + (n-1)(3.2)$

Step 3: Matching the options

Looking at the options:

1. $a_n = 7 + (n-1)3.2$: This matches our derived formula.
2. $a_n = 7(3.2)^{n-1}$: Incorrect, as this is a geometric sequence formula, not an arithmetic sequence.
3. $a_n = 3.2 + (n-1)7$: Incorrect, as it reverses the roles of the initial term and the common difference.
4. $a_n = a_{n-1} + (n-1)3.2$: Incorrect, as it does not represent the explicit formula but a recursive relationship.

Final Answer:

The correct choice is $a_n = 7 + (n-1)3.2$.

Would you like me to explain any of the steps in more detail?

Related Questions:

1. How is the explicit formula for an arithmetic sequence derived?
2. How do we distinguish between arithmetic and geometric sequences?
3. Can an arithmetic sequence have a common difference of 0?
4. How would the explicit formula change if the sequence started at $a_1 = 0$?
5. What is the relationship between the recursive and explicit formulas of a sequence?

Tip:

Always remember the formula for arithmetic sequences: $a_n = a_1 + (n-1)d$. It helps connect the initial term to the common difference.