The problem provided is about generating a sequence using a specific set of operations. The first number in the sequence is 6, and the next number is generated using the following steps:

1. Square the previous number.
2. Compute the sum of the digits of the squared result.
3. Subtract 2 from the sum obtained in step 2.

Given the first number is 6, let's walk through how we get the next numbers:

Example Calculation:

• Starting with 6:

  1. Square it: $6^2 = 36$
  2. Sum the digits of 36: $3 + 6 = 9$
  3. Subtract 2: $9 - 2 = 7$
  • So, the next number is 7.

• Starting with 7:

  1. Square it: $7^2 = 49$
  2. Sum the digits of 49: $4 + 9 = 13$
  3. Subtract 2: $13 - 2 = 11$
  • So, the next number is 11.

Now, you are asked to find the 2020th number in this sequence. Let's start by analyzing the sequence further and explore whether it cycles, which is a common pattern in such problems. I'll calculate the next few terms and see if a cycle emerges.

Let me compute this.The sequence generated is:

$6, 7, 11, 2$

It repeats after the fourth term, forming a cycle. This means that the sequence cycles every 4 terms. Given this, the 2020th term in the sequence can be found by calculating the position of 2020 modulo the cycle length (which is 4).

Let me compute that.The 2020th number in the sequence is 2.

Would you like further details on the process, or have any other questions?

Here are five related questions to deepen your understanding:

1. What is the general rule for identifying cycles in sequences?
2. How can we determine the length of a cycle in a mathematical sequence?
3. What are other types of sequences that exhibit periodic behavior?
4. Can you prove that the sequence will always cycle in 4 terms?
5. How would the sequence change if we modified the subtraction step in part (c)?

Tip: When solving problems involving sequences, detecting cycles early on can significantly reduce computational effort.