Next Number Please! Use inductive reasoning to predict the next number in the list and choose a proper conjecture. 1, 4, 9, 16, 25, 36, 49, ___

1. Break Down the Problem

The given sequence is: 1, 4, 9, 16, 25, 36, 49. We need to find the next number in this series.

2. Relevant Concepts

Observing the sequence, we can identify that these numbers are perfect squares:

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$
• $7^2 = 49$

3. Analysis and Detail

The pattern shows that the sequence consists of the squares of consecutive integers. Continuing this pattern, the next integer after 7 is 8, and therefore: $8^2 = 64$

4. Verify and Summarize

We have identified that the numbers in the sequence are $n^2$ where $n$ is a consecutive integer starting from 1. Thus, the next number which is $8^2$ confirms our conjecture.

Final Answer

Therefore, the next number in the sequence is 64.