Let's solve this step by step.

1. Initial Condition:

   • The number of tulips initially is $2n + 1$.
   • The number of daffodils is 12.

2. After Adding Tulips:

   • 3 more tulips are added, making the new total of tulips $(2n + 1) + 3 = 2n + 4$.

3. New Ratio Condition:

   • The new ratio of tulips to daffodils is 11:6.
   • This gives the equation: $\frac{2n + 4}{12} = \frac{11}{6}.$

4. Solve the Equation:

   • Cross-multiply to solve for $n$: $6(2n + 4) = 11 \times 12.$
   • This simplifies to: $12n + 24 = 132.$
   • Subtract 24 from both sides: $12n = 108.$
   • Divide by 12: $n = 9.$

Therefore, the value of $n$ is 9.

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions for further practice:

1. What would be the new ratio if 5 more tulips were added instead of 3?
2. If the ratio of tulips to daffodils was instead 5:3 after adding 3 tulips, what would be the value of $n$?
3. How many tulips and daffodils are there in total when $n = 9$?
4. What happens to the ratio if 2 daffodils are removed from the flowerbed?
5. If initially there were twice as many tulips as daffodils, what equation would you use to find $n$?

Tip: When solving ratio problems, setting up an equation that equates the two ratios is crucial for finding the unknown value.