To find the sequence of odd numbers that sum up to a given cube, we can use the following property of cubes:

The sum of the first $n$ odd numbers is $n^2$. For instance, the sum of the first 3 odd numbers (1, 3, 5) is $3^2 = 9$.

Similarly, for cubes: $n^3 = \sum_{k=1}^{n} (2k-1)$

This means the cube of $n$ is the sum of the first $n$ odd numbers, then the next $n$ odd numbers, and so on.

To clarify, let's solve for each case:

(a) 7

To find the sequence of odd numbers that sum to $7^3 = 343$:

• First, we need the sum of the first 7 odd numbers: $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$
• Then the sum of the next 7 odd numbers: $15 + 17 + 19 + 21 + 23 + 25 + 27 = 161$
• Finally, the sum of the next 7 odd numbers: $29 + 31 + 33 + 35 + 37 + 39 + 41 = 343$

Thus, the sum of these three sets gives: $49 + 161 + 343 = 343$

So, the odd numbers are $1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41$.

(b) 12

To find the sequence of odd numbers that sum to $12^3 = 1728$:

• First, we need the sum of the first 12 odd numbers: $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = 144$
• Then the sum of the next 12 odd numbers: $25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 + 41 + 43 + 45 + 47 = 432$
• Continue in this pattern until the total sum reaches 1728.

This means, the odd numbers to be added are the first 36 odd numbers.

(c) 4

To find the sequence of odd numbers that sum to $4^3 = 64$:

• First, we need the sum of the first 4 odd numbers: $1 + 3 + 5 + 7 = 16$
• Then the sum of the next 4 odd numbers: $9 + 11 + 13 + 15 = 48$

So, the sum of these two sets gives: $16 + 48 = 64$

The odd numbers are $1, 3, 5, 7, 9, 11, 13, 15$.

Do you want details or have any questions?

Here are some related questions for further exploration:

1. How can we generalize the formula for the sum of odd numbers to get any $n$-th cube?
2. What is the relationship between consecutive cubes and their sum of odd numbers?
3. How can we use the sum of odd numbers to solve other polynomial equations?
4. How does the property of odd numbers summing to cubes extend to higher powers?
5. What are the implications of this property in number theory?
6. Can we derive similar properties for sums of even numbers?
7. How can these sums be represented visually to enhance understanding?
8. How does this property connect with other mathematical sequences or series?

Tip: When solving mathematical problems, breaking them down into smaller, manageable parts can make the solution more apparent and easier to verify.