The reason I didn't multiply 4 and 13 directly is because the summation notation represents the sum of multiple terms, each of which is $4Y_i$ where $i$ ranges from 1 to 13.

Here’s why:

1. Sigma Notation Basics: The sigma (∑) notation is used to represent the sum of a sequence of terms. In your case, the expression $\sum_{i=1}^{13} 4Y_i$ means that for each value of $i$ from 1 to 13, you take the term $4Y_i$ and sum them all together.

2. Term-by-Term Addition: The notation implies that $4Y_1 + 4Y_2 + \cdots + 4Y_{13}$ are all added together. Multiplying 4 by 13 would not account for the summation of the individual $Y_i$ terms but rather would incorrectly assume that all the $Y_i$ values are equal to 1.

However, if all the $Y_i$ values were indeed equal to a constant $Y$, then the sum could be simplified. In that case, the expression would become:

$\sum_{i=1}^{13} 4Y = 4Y \times 13 = 52Y$

But in the general case, each $Y_i$ could be different, so the summation must be carried out individually over all terms.

Would you like more details or have further questions?

Here are some follow-up questions:

1. What happens if the sequence $Y_i$ has a known pattern (e.g., arithmetic or geometric)?
2. How would the summation change if $Y_i$ was replaced with a specific function of $i$?
3. Can you factor out constants from a summation, and under what circumstances?
4. What would the result be if $Y_i$ were all equal to 1?
5. How would you apply this summation in a practical scenario, like calculating the total cost of items?

Tip: Always double-check whether the terms inside the summation are constants or variables before simplifying the expression.