To complete the sequence 1, 2, 4, 8, 16, 32, . . . 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. $1 \times 2 = 2$
2. $2 \times 2 = 4$
3. $4 \times 2 = 8$
4. $8 \times 2 = 16$
5. $16 \times 2 = 32$

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

• $32 \times 2 = 64$
• $64 \times 2 = 128$
• $128 \times 2 = 256$

So, the next three terms are 64, 128, 256.

Deductive Reasoning:

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

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

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

• For $n = 7$: $\text{Term}_7 = 2^{(7-1)} = 2^6 = 64$
• For $n = 8$: $\text{Term}_8 = 2^{(8-1)} = 2^7 = 128$
• For $n = 9$: $\text{Term}_9 = 2^{(9-1)} = 2^8 = 256$

So, using deductive reasoning, the next three terms are 64, 128, 256.

Summary:

Both inductive and deductive reasoning lead to the conclusion that the next three terms in the sequence are 64, 128, 256.

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the next term after 256 in the sequence?
2. Can you identify a different sequence where each term is multiplied by a number other than 2?
3. How would the sequence change if the starting term was 3 instead of 1?
4. What is the 10th term in the original sequence?
5. How can this sequence be represented using a recursive formula?

Tip: Deductive reasoning is often more powerful when you know the general rule governing a pattern, while inductive reasoning is useful when you're identifying the rule based on observed data.