The problem provided asks to express the two sums given as a single summation. The two sums are:

$\sum_{k=1}^{100} (2k - 1)^2 + \sum_{k=0}^{99} (2k - 1)^2$

Let's break this down and analyze the expression step by step.

Step 1: Check for overlapping indices

The first sum starts at $k = 1$ and ends at $k = 100$. The second sum starts at $k = 0$ and ends at $k = 99$. The terms in both summations are $(2k - 1)^2$.

• For the first sum:

  • At $k = 1$, the term is $(2 \times 1 - 1)^2 = 1^2$.
  • At $k = 100$, the term is $(2 \times 100 - 1)^2 = 199^2$.

• For the second sum:

  • At $k = 0$, the term is $(2 \times 0 - 1)^2 = (-1)^2 = 1$.
  • At $k = 99$, the term is $(2 \times 99 - 1)^2 = 197^2$.

Step 2: Combine both sums

To combine the two sums into a single summation, observe that they almost cover the same terms, except the first sum has the extra term at $k = 100$.

Therefore, we can express the combined sum as:

$\sum_{k=0}^{99} (2k - 1)^2 + (2 \times 100 - 1)^2$

This is equivalent to:

$\sum_{k=0}^{100} (2k - 1)^2$

Final Answer:

Thus, the two sums combined into a single summation is:

$\sum_{k=0}^{100} (2k - 1)^2$

This is the final simplified expression.

Would you like more details on how to simplify summations or any related topics?

Here are 5 related questions:

1. How do you simplify other types of sums with different indices?
2. What is the general approach to combine summations with different limits?
3. How can this sum be expanded to higher powers or different sequences?
4. What are common mistakes to avoid when handling summation indices?
5. How can the properties of even and odd numbers affect summations?

Tip: When combining sums, always check for overlapping terms and adjust the limits accordingly.