1. **State the problem:** We are given the sequence $1, 3, 4, 7, 11, \ldots$ and need to find a recursive rule for it and then write the next two terms. 2. **Analyze the sequence:** The first two terms are given as $a_1 = 1$ and $a_2 = 3$. 3. **Find the recursive rule:** Look at how terms relate to previous terms. Calculate differences: $$3 - 1 = 2$$ $$4 - 3 = 1$$ $$7 - 4 = 3$$ $$11 - 7 = 4$$ Notice the pattern in differences resembles the sequence itself shifted. Try the recursive formula: $$a_n = a_{n-1} + a_{n-2}$$ Check for $n=3$: $$a_3 = a_2 + a_1 = 3 + 1 = 4$$ (matches) Check for $n=4$: $$a_4 = a_3 + a_2 = 4 + 3 = 7$$ (matches) Check for $n=5$: $$a_5 = a_4 + a_3 = 7 + 4 = 11$$ (matches) 4. **Write the recursive rule:** $$a_1 = 1, \quad a_2 = 3, \quad a_n = a_{n-1} + a_{n-2} \text{ for } n \geq 3$$ 5. **Find the next two terms:** $$a_6 = a_5 + a_4 = 11 + 7 = 18$$ $$a_7 = a_6 + a_5 = 18 + 11 = 29$$ **Final answer:** Recursive rule: $a_1 = 1$, $a_2 = 3$, $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$ Next two terms: $18$ and $29$