<Question>
Let \[f(n) =
\begin{cases}
4n+3 &\text{if }n<a, \\
7n-12 &\text{if }n\ge{a}.
\end{cases}
\]Find $a$ if the graph of $y=f(n)$ is continuous.
</Question>

<think>
<Plan_1>
Okay, so I need to find the value of 'a' that makes the function f(n) continuous. Let me start by recalling what continuity means for a function at a point. If a function is continuous at a point, then the left-hand limit and the right-hand limit at that point must be equal, and they must also equal the function's value at that point. In this case, the function f(n) is piecewise-defined, with different expressions depending on whether n is less than a or greater than or equal to a. So the potential point of discontinuity is at n = a. Therefore, to ensure continuity, the limit as n approaches a from below (using the 4n + 3 part) should equal the limit as n approaches a from above (using the 7n - 12 part), and both should equal f(a).

Let me write that down mathematically. The left-hand limit as n approaches a is:

limₙ→a⁻ f(n) = 4a + 3

The right-hand limit as n approaches a is:

limₙ→a⁺ f(n) = 7a - 12

For the function to be continuous at n = a, these two limits must be equal:

<prompt_1.1> Set up the equation for the left-hand limit and the right-hand limit and solve for 'a'. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
4a + 3 = 7a - 12

Now I need to solve this equation for a. Let me subtract 4a from both sides:

3 = 3a - 12

Then, add 12 to both sides:

15 = 3a

Divide both sides by 3:

a = 5
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, let me check that again. If a is 5, then for n < 5, the function is 4n + 3, and for n ≥ 5, it's 7n - 12. Let me verify the continuity at n = 5.

<prompt_2.1> Verify the solution by checking the function values around n = 5. </prompt_2.1>

But wait, is there any other condition I need to check? For example, the function is defined piecewise, but is there anything else that could affect continuity?

<prompt_2.2> Check if there are any other conditions or aspects that could affect the continuity of the function. </prompt_2.2>

Let me just make sure I didn't make any calculation mistakes. Let's re-solve the equation:

<prompt_2.3> Re-solve the equation to confirm the solution for 'a'. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
Calculating the left-hand limit as n approaches 5: 4*5 + 3 = 20 + 3 = 23.

Calculating the right-hand limit as n approaches 5: 7*5 - 12 = 35 - 12 = 23.

And f(5) is 7*5 - 12 = 23, so the function value at 5 is also 23. Therefore, all three are equal, so it's continuous at a = 5.
</execution_2.1>

<execution_2.2>
The function is defined as two linear pieces. The only point where continuity might not hold is at the boundary n = a. So as long as the left and right limits at a are equal, the function will be continuous there. Since we found a = 5, that should be the answer.
</execution_2.2>

<execution_2.3>
Original equation from limits:

4a + 3 = 7a - 12

Subtract 4a:

3 = 3a - 12

Add 12:

15 = 3a

Divide by 3:

a = 5
</execution_2.3>
</Execution_2>

<Final_answer>
Therefore, the value of $a$ that ensures the graph of $y=f(n)$ is continuous is $\boxed{5}$.
</Final_answer>
</think>