List of sequences with their respective formulas.

It looks like you've uploaded a list of sequences defined by formulas for the $ n $-th term $ a_n $. However, since I can't view images directly, you’ve provided the text of the problem. Based on what you've written, it appears you're being asked to analyze or evaluate these sequences — possibly to:

- Find specific terms (e.g., $ a_1, a_2, \dots $),
- Determine whether they are arithmetic, geometric, or neither,
- Identify patterns,
- Or find limits as $ n \to \infty $.

Since no specific instruction is given in your message (like "find the first 5 terms", "classify each sequence", or "evaluate the limit"), I’ll assume the most common task: Analyze each sequence, including computing the first few terms and determining its behavior (e.g., convergence, type of sequence).

Let’s go through each one step by step.

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a) $ a_n = -5 $

This is a constant sequence.

- All terms are $ -5 $.
- $ a_1 = -5, a_2 = -5, \dots $
- Type: Constant
- Limit: $ \lim_{n \to \infty} a_n = -5 $

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b) $ a_n = \frac{n-1}{n} $

Simplify:
$$
a_n = 1 - \frac{1}{n}
$$

- $ a_1 = \frac{0}{1} = 0 $
- $ a_2 = \frac{1}{2} = 0.5 $
- $ a_3 = \frac{2}{3} \approx 0.666 $
- $ a_4 = \frac{3}{4} = 0.75 $
- As $ n \to \infty $, $ \frac{1}{n} \to 0 $, so $ a_n \to 1 $

- Behavior: Increases toward 1
- Limit: $ \lim_{n \to \infty} a_n = 1 $
- Type: Not arithmetic or geometric; rational function

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c) $ a_n = 3n $

- $ a_1 = 3, a_2 = 6, a_3 = 9, \dots $
- This is an arithmetic sequence with common difference 3.
- General form: $ a_n = 3n $
- Limit: $ \lim_{n \to \infty} a_n = \infty $

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d) $ a_n = \left( \frac{2}{3} \right)^{n-1} $

- This is a geometric sequence with ratio $ r = \frac{2}{3} < 1 $
- $ a_1 = \left(\frac{2}{3}\right)^0 = 1 $
- $ a_2 = \frac{2}{3} $
- $ a_3 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} $
- $ a_4 = \left(\frac{2}{3}\right)^3 = \frac{8}{27} $
- Limit: Since $ |r| < 1 $, $ \lim_{n \to \infty} a_n = 0 $

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e) $ a_n = \left( \frac{2}{3} \right)^{1-n} $

Rewrite:
$$
a_n = \left( \frac{2}{3} \right)^{-(n-1)} = \left( \frac{3}{2} \right)^{n-1}
$$

So this is a geometric sequence with ratio $ r = \frac{3}{2} > 1 $

- $ a_1 = \left(\frac{2}{3}\right)^0 = 1 $
- $ a_2 = \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} = 1.5 $
- $ a_3 = \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = 2.25 $
- $ a_4 = \left(\frac{3}{2}\right)^3 = 3.375 $
- Grows exponentially
- Limit: $ \lim_{n \to \infty} a_n = \infty $

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f) $ a_n = \frac{2n}{7} $

- $ a_1 = \frac{2}{7}, a_2 = \frac{4}{7}, a_3 = \frac{6}{7}, \dots $
- This is an arithmetic sequence with common difference $ \frac{2}{7} $
- Limit: $ \lim_{n \to \infty} a_n = \infty $

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g) $ a_n = n^n $

- $ a_1 = 1^1 = 1 $
- $ a_2 = 2^2 = 4 $
- $ a_3 = 3^3 = 27 $
- $ a_4 = 4^4 = 256 $
- Extremely fast-growing
- Limit: $ \lim_{n \to \infty} a_n = \infty $
- Note: This grows faster than exponential or factorial

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h) $ a_n = 4 \times 3^{1-n} $

Rewrite:
$$
a_n = 4 \cdot 3 \cdot 3^{-n} = 12 \cdot \left(\frac{1}{3}\right)^n
$$
or better:
$$
a_n = 4 \cdot 3^{1-n} = \frac{4}{3^{n-1}}
$$

- $ a_1 = 4 \cdot 3^{0} = 4 $
- $ a_2 = 4 \cdot 3^{-1} = \frac{4}{3} \approx 1.333 $
- $ a_3 = 4 \cdot 3^{-2} = \frac{4}{9} \approx 0.444 $
- $ a_4 = \frac{4}{27} \approx 0.148 $
- Geometric decay, ratio $ \frac{1}{3} $
- Limit: $ \lim_{n \to \infty} a_n = 0 $

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i) $ a_n = (n+1)^3 $

- $ a_1 = 2^3 = 8 $
- $ a_2 = 3^3 = 27 $
- $ a_3 = 4^3 = 64 $
- Cubic growth
- Limit: $ \lim_{n \to \infty} a_n = \infty $
- Not arithmetic or geometric

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j) $ a_n = \frac{(n+1)^2}{2} $

- $ a_1 = \frac{4}{2} = 2 $
- $ a_2 = \frac{9}{2} = 4.5 $
- $ a_3 = \frac{16}{2} = 8 $
- $ a_4 = \frac{25}{2} = 12.5 $
- Quadratic growth
- Limit: $ \infty $
- Not arithmetic or geometric

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k) $ a_n = (-2)^n $

- $ a_1 = -2 $
- $ a_2 = 4 $
- $ a_3 = -8 $
- $ a_4 = 16 $
- Alternating sign, magnitude doubles each time
- Geometric sequence with $ r = -2 $
- Since $ |r| > 1 $, it diverges (oscillates and grows)
- Limit: Does not exist (no limit)

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l) $ a_n = n(n-1) $

- $ a_1 = 1(0) = 0 $
- $ a_2 = 2(1) = 2 $
- $ a_3 = 3(2) = 6 $
- $ a_4 = 4(3) = 12 $
- $ a_5 = 5(4) = 20 $
- Quadratic growth: $ a_n = n^2 - n $
- Limit: $ \infty $
- Not arithmetic or geometric

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## ✔ Summary Table

| Letter | Formula | Type | First Few Terms | Limit |
|--------|--------------------------|--------------------|-------------------------|-------|
| a) | $ a_n = -5 $ | Constant | -5, -5, -5, ... | -5 |
| b) | $ a_n = \frac{n-1}{n} $| Rational | 0, 0.5, 0.666..., →1 | 1 |
| c) | $ a_n = 3n $ | Arithmetic | 3, 6, 9, 12, ... | ∞ |
| d) | $ a_n = (2/3)^{n-1} $ | Geometric (r<1) | 1, 2/3, 4/9, ... →0 | 0 |
| e) | $ a_n = (2/3)^{1-n} $ | Geometric (r>1) | 1, 1.5, 2.25, ... →∞ | ∞ |
| f) | $ a_n = 2n/7 $ | Arithmetic | 2/7, 4/7, 6/7, ... | ∞ |
| g) | $ a_n = n^n $ | Fast-growing | 1, 4, 27, 256, ... | ∞ |
| h) | $ a_n = 4 \cdot 3^{1-n} $ | Geometric (r<1) | 4, 4/3, 4/9, ... →0 | 0 |
| i) | $ a_n = (n+1)^3 $ | Cubic | 8, 27, 64, ... | ∞ |
| j) | $ a_n = \frac{(n+1)^2}{2} $ | Quadratic | 2, 4.5, 8, 12.5, ... | ∞ |
| k) | $ a_n = (-2)^n $ | Geometric (r=-2) | -2, 4, -8, 16, ... | DNE |
| l) | $ a_n = n(n-1) $ | Quadratic | 0, 2, 6, 12, 20, ... | ∞ |

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If you had a specific question (e.g., “Which sequences converge?”, “Find the sum of the first 5 terms”, or “Classify each”), let me know and I can tailor the answer further!