To complete the sequence 1, 3, 5, 7, 9, 11, . . . 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. $1 + 2 = 3$
2. $3 + 2 = 5$
3. $5 + 2 = 7$
4. $7 + 2 = 9$
5. $9 + 2 = 11$

The pattern shows that each term is obtained by adding 2 to the previous term. Based on this pattern, we can inductively predict that the next terms in the sequence would be:

• $11 + 2 = 13$
• $13 + 2 = 15$
• $15 + 2 = 17$

So, the next three terms are 13, 15, 17.

Deductive Reasoning:

Deductive reasoning involves applying a general rule to a specific case. The general rule in this sequence can be expressed as:

$\text{Term}_n = 2n - 1$

Where $n$ is the position of the term in the sequence.

• For $n = 7$: $\text{Term}_7 = 2(7) - 1 = 14 - 1 = 13$
• For $n = 8$: $\text{Term}_8 = 2(8) - 1 = 16 - 1 = 15$
• For $n = 9$: $\text{Term}_9 = 2(9) - 1 = 18 - 1 = 17$

So, using deductive reasoning, the next three terms are 13, 15, 17.

Summary:

Both inductive and deductive reasoning lead to the conclusion that the next three terms in the sequence are 13, 15, 17.

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 the sequence?
2. How would the sequence change if the starting term was 2 instead of 1?
3. Can you create a sequence where each term is the result of adding 3 instead of 2 to the previous term?
4. How can this sequence be represented using a recursive formula?
5. What is the general form for an arithmetic sequence like this one?

Tip: In an arithmetic sequence, the difference between consecutive terms is constant. This difference is called the "common difference."