We are given the sequence $Q$ where:

• $Q_1 = 1$
• For $n \geq 2$, $Q_n$ is the absolute value of the difference between the $n$-th smallest positive perfect cube and the $(n-1)$-th smallest positive perfect cube.

Step 1: List of Perfect Cubes

The smallest positive perfect cubes are: $1^3 = 1, \quad 2^3 = 8, \quad 3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125, \quad 6^3 = 216, \quad 7^3 = 343$

Step 2: Differences Between Consecutive Perfect Cubes

Now, let's compute the absolute value of the difference between consecutive cubes:

• $Q_2 = |8 - 1| = 7$
• $Q_3 = |27 - 8| = 19$
• $Q_4 = |64 - 27| = 37$
• $Q_5 = |125 - 64| = 61$
• $Q_6 = |216 - 125| = 91$
• $Q_7 = |343 - 216| = 127$

Step 3: Sum of the First Seven Terms

Now, we add the first seven terms of the sequence: $Q_1 + Q_2 + Q_3 + Q_4 + Q_5 + Q_6 + Q_7 = 1 + 7 + 19 + 37 + 61 + 91 + 127$ $= 343$

Thus, the sum of the first seven terms in sequence $Q$ is $\boxed{343}$.

Would you like further details or clarifications?

Here are 5 related questions to further explore the topic:

1. What is the general formula for the difference between consecutive perfect cubes?
2. How does the growth rate of perfect cubes compare to perfect squares?
3. What is the sum of the first 10 terms in sequence $Q$?
4. How can perfect cubes be represented algebraically?
5. What are other sequences that involve differences of powers?

Tip: The difference between two consecutive cubes follows a quadratic pattern, increasing more rapidly as the numbers grow.