2. Find the sequences {an + bn} and {anbn}, where the sequences {an} and {bn} are defined by an= 2n2 ∀n ∈ **ℕ**, bn = 1/n ∀n ∈ **ℕ**.

Given the sequences \(\{a_n\}\) and \(\{b_n\}\), where:

- \(a_n = 2n^2\), for all \(n \in \mathbb{N}\),
- \(b_n = \frac{1}{n}\), for all \(n \in \mathbb{N}\),

we are tasked with finding the sequences \(\{a_n + b_n\}\) and \(\{a_n b_n\}\).

### Sequence 1: \(\{a_n + b_n\}\)

To find the sequence \(\{a_n + b_n\}\), we simply add the given formulas for \(a_n\) and \(b_n\):

\[
a_n + b_n = 2n^2 + \frac{1}{n}
\]

Thus, the sequence \(\{a_n + b_n\}\) is:

\[
\{a_n + b_n\} = \left\{2n^2 + \frac{1}{n}\right\} \quad \text{for all } n \in \mathbb{N}
\]

### Sequence 2: \(\{a_n b_n\}\)

To find the sequence \(\{a_n b_n\}\), we multiply the given formulas for \(a_n\) and \(b_n\):

\[
a_n b_n = \left(2n^2\right) \cdot \left(\frac{1}{n}\right) = 2n
\]

Thus, the sequence \(\{a_n b_n\}\) is:

\[
\{a_n b_n\} = \{2n\} \quad \text{for all } n \in \mathbb{N}
\]

### Summary

- \(\{a_n + b_n\} = \left\{2n^2 + \frac{1}{n}\right\}\)
- \(\{a_n b_n\} = \{2n\}\)

Would you like further details or have any questions?

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Here are 5 related questions:
1. How does the behavior of the sequence \(\{a_n + b_n\}\) change as \(n\) grows large?
2. Can you express \(\{a_n + b_n\}\) in terms of limits as \(n \to \infty\)?
3. How would the sequence change if \(b_n = \frac{1}{n^2}\) instead?
4. What is the general formula for the product of two sequences \(\{a_n\}\) and \(\{b_n\}\)?
5. How would these sequences change if \(a_n\) were defined as \(a_n = 3n^2 + 1\)?

**Tip**: When dealing with sequences, analyzing their behavior for large \(n\) often gives useful insights into their long-term behavior.

Find the sequences {a_n + b_n} and {a_n b_n}, where the sequences {a_n} and {b_n} are defined by a_n = 2n^2 ∀n ∈ ℕ, b_n = 1/n ∀n ∈ ℕ.

Explore how to find the sequences {a_n + b_n} and {a_n b_n} for a_n = 2n^2 and b_n = 1/n. This problem provides a detailed solution for both addition and multiplication of sequences. Suitable for high school students in Grades 9-12.

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