To find a recursive formula for the sequence $-12, 3, 18, 33$ and calculate the next term, let's analyze it step by step:

Step 1: Identify the differences between terms

$\text{Second term} - \text{First term} = 3 - (-12) = 15$ $\text{Third term} - \text{Second term} = 18 - 3 = 15$ $\text{Fourth term} - \text{Third term} = 33 - 18 = 15$ The sequence has a constant difference of 15, so it's an arithmetic sequence.

Step 2: Recursive formula for an arithmetic sequence

For an arithmetic sequence with a constant difference $d$: $a_{n+1} = a_n + d$ Here, $d = 15$. The recursive formula becomes: $a_{n+1} = a_n + 15 \quad \text{with } a_1 = -12$

Step 3: Find the next term

Using the formula: $a_5 = a_4 + 15 = 33 + 15 = 48$

Final Answer:

• The next term is $48$.
• The recursive formula is: $a_{n+1} = a_n + 15, \quad a_1 = -12$

Would you like me to elaborate on deriving this or explore other types of formulas?

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5 Related Questions:

1. How would the explicit formula for this arithmetic sequence look?
2. Can we generalize the sum of the first $n$ terms for this sequence?
3. How would the sequence behave if the difference $d$ was not constant?
4. What happens if we start from a different first term but maintain the same difference?
5. Can you find recursive formulas for non-arithmetic sequences?

Tip:

For arithmetic sequences, the recursive formula is always of the form $a_{n+1} = a_n + d$, and the explicit formula involves both the first term and the difference!