find the first three terms of the sequence defined by     1. an=3n-5     2. an=n(n+1)     3. 2n=2^n-3

Let's find the first three terms for each of the sequences given:

### 1. Sequence \( a_n = 3n - 5 \)
For the first three terms, plug in \( n = 1, 2, 3 \) into the formula:

- \( a_1 = 3(1) - 5 = 3 - 5 = -2 \)
- \( a_2 = 3(2) - 5 = 6 - 5 = 1 \)
- \( a_3 = 3(3) - 5 = 9 - 5 = 4 \)

So, the first three terms are: **\(-2, 1, 4\)**

### 2. Sequence \( a_n = n(n + 1) \)
Again, for the first three terms, plug in \( n = 1, 2, 3 \):

- \( a_1 = 1(1 + 1) = 1 \times 2 = 2 \)
- \( a_2 = 2(2 + 1) = 2 \times 3 = 6 \)
- \( a_3 = 3(3 + 1) = 3 \times 4 = 12 \)

So, the first three terms are: **\(2, 6, 12\)**

### 3. Sequence defined by \( 2n = 2^n - 3 \)
Here, you need to solve the equation for \( n \) to find each term \( a_n \). The sequence is defined implicitly:

- For \( n = 1 \): \( 2(1) = 2^1 - 3 \Rightarrow 2 = 2 - 3 \) (not true)
- For \( n = 2 \): \( 2(2) = 2^2 - 3 \Rightarrow 4 = 4 - 3 = 1 \) (not true)
- For \( n = 3 \): \( 2(3) = 2^3 - 3 \Rightarrow 6 = 8 - 3 = 5 \) (not true)

It appears the equation \( 2n = 2^n - 3 \) does not produce a valid sequence for the first few values of \( n \). This is a non-standard sequence, and might have no solutions for small \( n \).

Therefore, the first two sequences produce valid terms:
1. **\(-2, 1, 4\)**
2. **\(2, 6, 12\)**

The third sequence requires a different approach or clarification.

Would you like further details on any of these? 

Here are five related questions you might consider:
1. What happens if we try a different formula for the third sequence?
2. How can we derive a formula if given a non-standard sequence?
3. Can we find a recursive definition for the first or second sequences?
4. What would be the first three terms if the formula in sequence 1 was modified?
5. How does the pattern change if we alter the starting index \( n \) for these sequences?

**Tip:** When working with sequences, always ensure that the formula works for all given values of \( n \) before assuming it's a standard sequence.

Explore mathematical sequences defined by formulas such as an = 3n - 5 and an = n(n + 1), and analyze their patterns. Discover the solution process and challenges in defining non-standard sequences like 2n = 2^n - 3. Suitable for high school students.

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