Given the problem, let's outline and solve it systematically.

Given Information:

1. Type of Series: Engineering series (usually implies a geometric series in engineering).
2. Even Number of Terms: $q = 4$, meaning there are at least 4 terms.
3. First Term: $a = 2$.
4. Sum of the Last Three Terms: 2688.

Step 1: General Formulation

In a geometric series:

• The terms are expressed as: $a, ar, ar^2, \ldots, ar^{n-1}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Since we know the sum of the last three terms, let's express them:

• Last Three Terms (assuming total number of terms $n$):
  • Third last term: $ar^{n-3}$
  • Second last term: $ar^{n-2}$
  • Last term: $ar^{n-1}$

The sum of these last three terms is given by: $ar^{n-3} + ar^{n-2} + ar^{n-1} = 2688$

Step 2: Substitute Known Values

We know $a = 2$. Substituting $a$ gives: $2r^{n-3} + 2r^{n-2} + 2r^{n-1} = 2688$ Dividing both sides by 2 simplifies this to: $r^{n-3} + r^{n-2} + r^{n-1} = 1344$

Step 3: Solve for $n$ Using Known Values of Terms

Since $q = 4$, we can try calculating $n$ under the assumption that we have a four-term geometric series. Given the sum relationship, we would proceed with possible values for $r$ that satisfy the series pattern and calculate further.

Would you like to proceed by testing specific values for $r$ to simplify further, or is there additional information on the common ratio $r$ to refine our calculations?