To find the first three terms of the sequence $a_n = \frac{n+2}{3n^2}$, we substitute $n = 1$, $n = 2$, and $n = 3$ into the given formula.

1. For $n = 1$: $a_1 = \frac{1+2}{3(1^2)} = \frac{3}{3} = 1$

2. For $n = 2$: $a_2 = \frac{2+2}{3(2^2)} = \frac{4}{12} = \frac{1}{3}$

3. For $n = 3$: $a_3 = \frac{3+2}{3(3^2)} = \frac{5}{27}$

The first three terms are:

$1, \frac{1}{3}, \frac{5}{27}$

Would you like me to explain further or calculate additional terms?

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Related Questions:

1. What is the behavior of the sequence as $n \to \infty$?
2. Is this sequence monotonic (increasing or decreasing)?
3. How does this sequence relate to harmonic or geometric sequences?
4. Can we find a general formula for the sum of the first $n$ terms?
5. How would the sequence change if the denominator were $3n^3$ instead?

Tip:

Always simplify the sequence term-by-term calculation to ensure clarity and avoid errors!