To derive a recursive formula from an explicit formula, you typically aim to express $a_n$ in terms of $a_{n-1}$ (or occasionally earlier terms).

Explicit Formula (Given):

$a_n = 2 + 5n$

Step-by-Step Derivation of Recursive Formula:

1. Start with the explicit formula: $a_n = 2 + 5n$

2. Find the relationship between successive terms: Subtract $a_{n-1}$ from $a_n$: $a_n - a_{n-1} = (2 + 5n) - (2 + 5(n-1))$

3. Simplify the difference: $a_n - a_{n-1} = 2 + 5n - 2 - 5n + 5 = 5$ Thus, $a_n = a_{n-1} + 5$.

4. Define the initial condition: From the explicit formula, when $n = 1$: $a_1 = 2 + 5(1) = 7$

Recursive Formula:

The recursive formula is: $a_n = a_{n-1} + 5, \quad \text{with } a_1 = 7$

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Interpretation:

• Each term increases by 5 from the previous one, starting at 7.

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Would you like further clarification or examples? Here are some follow-up questions:

1. How do you convert other explicit formulas into recursive ones?
2. Can we derive the explicit formula back from a recursive one?
3. What are examples of sequences where both recursive and explicit formulas are used?
4. How does this method extend to quadratic or non-linear sequences?
5. Can you explore sequences with different initial conditions?

Tip: When working with arithmetic sequences, the common difference directly gives the increment for the recursive formula.