Let’s analyze each sequence to determine the pattern and find the next number.

(a) $1, 8, 27, \dots$

This sequence is made up of cubes of integers: $1^3 = 1,\; 2^3 = 8,\; 3^3 = 27,\; \dots$ The next number is $4^3 = 64$.

(b) $1, 2, 4, 7, 11, \dots$

Here, the differences between consecutive terms are: $2 - 1 = 1,\; 4 - 2 = 2,\; 7 - 4 = 3,\; 11 - 7 = 4$ The differences increase by 1 each time. So, the next difference is $5$, and the next term is: $11 + 5 = 16$

(c) $1, 4, 9, 25, \dots$

This sequence is made up of squares of integers: $1^2 = 1,\; 2^2 = 4,\; 3^2 = 9,\; 5^2 = 25$ Notice the pattern skips $4^2$. The next number is $6^2 = 36$.

Final Answers:

• (a): $64$
• (b): $16$
• (c): $36$

Would you like further details or explanations?
Here are 5 related questions for practice:

1. What is the 10th term of the sequence $1, 8, 27, \dots$?
2. Find the sum of the first 5 terms of $1, 2, 4, 7, \dots$.
3. What would the 7th term of $1, 4, 9, 25, \dots$ be?
4. How can you identify a cubic sequence quickly?
5. Derive a formula for the $n$-th term of $1, 2, 4, 7, 11, \dots$.

Tip: Always check the differences between terms or powers of integers to uncover hidden patterns!