A. Use Inductive and Deductive reasoning to complete the following, provide three terms
1, 11, 22, 34, 47, . . .

To complete the sequence 1, 11, 22, 34, 47, . . . using inductive and deductive reasoning, let's analyze the pattern and predict the next terms.

### Inductive Reasoning:
Inductive reasoning involves identifying patterns from specific examples and making a generalization. Observing the given sequence:

1. \(11 - 1 = 10\) (difference between the 1st and 2nd terms)
2. \(22 - 11 = 11\) (difference between the 2nd and 3rd terms)
3. \(34 - 22 = 12\) (difference between the 3rd and 4th terms)
4. \(47 - 34 = 13\) (difference between the 4th and 5th terms)

The pattern shows that the difference between consecutive terms increases by 1 each time. Based on this pattern, we can inductively predict that the next terms in the sequence would have differences of 14, 15, and 16 from their preceding terms:

- \(47 + 14 = 61\)
- \(61 + 15 = 76\)
- \(76 + 16 = 92\)

So, the next three terms are **61, 76, 92**.

### Deductive Reasoning:
Deductive reasoning involves applying a general rule to a specific case. Given that the difference between terms is increasing by 1, the rule for the nth term in the sequence can be expressed as:

\[
\text{Term}_n = \text{Term}_{n-1} + (9 + n-1)
\]

Where each term is calculated by adding the increasing difference to the previous term:

- For \(n = 6\): \(\text{Term}_6 = 47 + 14 = 61\)
- For \(n = 7\): \(\text{Term}_7 = 61 + 15 = 76\)
- For \(n = 8\): \(\text{Term}_8 = 76 + 16 = 92\)

So, using deductive reasoning, the next three terms are **61, 76, 92**.

### Summary:
Both inductive and deductive reasoning lead to the conclusion that the next three terms in the sequence are **61, 76, 92**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is the 10th term in this sequence?
2. Can you find a general formula for the nth term of this sequence?
3. How would the sequence change if the first term was different?
4. What is the sum of the first 10 terms of this sequence?
5. Can you create a similar sequence where the difference increases by 2 instead of 1?

**Tip:** When identifying patterns in a sequence, look for changes in differences or ratios between terms to uncover potential arithmetic or geometric progressions.

Explore how to use inductive and deductive reasoning to complete the sequence 1, 11, 22, 34, 47 with detailed explanations and predictions. Learn about arithmetic sequences and the principles of inductive and deductive logic. Suitable for advanced high school students.

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