We observe that the sum follows a clear pattern: each sum involves a series of numbers starting from 1, increasing sequentially to some value, then decreasing back to 1. The pattern in the sums of the first few examples is as follows:

• $1 = 1^2$
• $1 + 2 + 1 = 2^2 = 4$
• $1 + 2 + 3 + 2 + 1 = 3^2 = 9$
• $1 + 2 + 3 + 4 + 3 + 2 + 1 = 4^2 = 16$
• $1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 5^2 = 25$

From this, we infer that the pattern is based on the square of the highest number in the sequence. For the given sequence:

$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1$

The highest number is 10, so the sum follows the pattern $10^2 = 100$.

Thus, the sum of this sequence is 100.

Answer: 100

Would you like more details or further clarifications on this pattern?

Related Questions:

1. How do you prove that the sum of numbers from 1 to n and back down is always a perfect square?
2. What would the sum be if the sequence reached a maximum of 12 instead of 10?
3. How can this pattern be applied to other types of arithmetic sequences?
4. Can this pattern be generalized for sequences that increase and then decrease in other mathematical contexts?
5. How can you derive the sum of an arithmetic series more generally?

Tip:

For sequences that increase and decrease symmetrically, observe the highest point in the sequence, and use its square as the result!