Let's analyze each statement one by one:

A. The tenth term of the sequence 4, 7, 10, 13, … is 30.

This is an arithmetic sequence where the first term $a_1 = 4$ and the common difference $d = 3$. The formula for the nth term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) d$

For $n = 10$, we have:

$a_{10} = 4 + (10 - 1) \cdot 3 = 4 + 9 \cdot 3 = 4 + 27 = 31$

So, the tenth term is 31, not 30. A is false.

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**B. $a_n = 11 + 6n$ and (a_n = \begin{cases}

11 & \text{if } n = 1 \ a_{n-1} + 6n & \text{if } n > 1 \end{cases}) both define the same sequence.**

Let's compare both definitions.

1. For $a_n = 11 + 6n$:

   For $n = 1$: $a_1 = 11 + 6 \times 1 = 17$
   For $n = 2$: $a_2 = 11 + 6 \times 2 = 23$
   For $n = 3$: $a_3 = 11 + 6 \times 3 = 29$

2. For the recursive sequence (a_n = \begin{cases} 11 & \text{if } n = 1 \ a_{n-1} + 6n & \text{if } n > 1 \end{cases}):

   For $n = 1$: $a_1 = 11$
   For $n = 2$: $a_2 = a_1 + 6 \times 2 = 11 + 12 = 23$
   For $n = 3$: $a_3 = a_2 + 6 \times 3 = 23 + 18 = 41$

Clearly, the sequences are different. So, B is false.

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C. The sum of the first 12 terms of the arithmetic series with $a_1 = −5$ and $a_{12} = 232$ is 39.

The formula for the sum of the first $n$ terms of an arithmetic series is:

$S_n = \frac{n}{2} (a_1 + a_n)$

For $n = 12$, $a_1 = -5$, and $a_{12} = 232$:

$S_{12} = \frac{12}{2} (-5 + 232) = 6 \times 227 = 1362$

The sum is 1362, not 39. So, C is false.

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D. $\sum_{i=1}^{20} (3i - 15) = 450$

The sum can be rewritten as:

$\sum_{i=1}^{20} (3i - 15) = \sum_{i=1}^{20} 3i - \sum_{i=1}^{20} 15$

The sum of the first 20 terms of $3i$ is:

$\sum_{i=1}^{20} 3i = 3 \sum_{i=1}^{20} i = 3 \times \frac{20(21)}{2} = 3 \times 210 = 630$

The sum of $15$ repeated 20 times is:

$\sum_{i=1}^{20} 15 = 15 \times 20 = 300$

Now, subtract:

$630 - 300 = 330$

So, the sum is 330, not 450. D is false.

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Conclusion:

All four statements are false.

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Do you want further details or have any questions?

Here are 5 related questions to explore:

1. How do you find the sum of an arithmetic sequence?
2. What is the difference between an explicit and recursive formula for sequences?
3. How do you calculate the nth term of an arithmetic sequence?
4. How can you decompose summations like $\sum_{i=1}^{n} (ai + b)$?
5. How do you identify if two sequences are equivalent?

Tip: When working with summations, break them into simpler parts to handle constants and linear terms separately.