Hence, or otherwise, write \( -4+3+10+\ldots+486 \) in sigma notation.

To express the series \( -4 + 3 + 10 + \ldots + 486 \) in sigma notation, we first need to identify the pattern or rule governing this series. Let's observe the sequence: - The first term \( a_1 = -4 \) - The second term \( a_2 = 3 \) - The third term \( a_3 = 10 \) To find the common difference, we calculate: \( a_2 - a_1 = 3 - (-4) = 7 \) \( a_3 - a_2 = 10 - 3 = 7 \) Each term increases by 7. Thus, the sequence can be described by the formula for the \( n \)-th term: \[ a_n = -4 + 7(n-1) \] Simplifying this gives us: \[ a_n = 7n - 11 \] Next, we need to determine how many terms there are in the sequence that ends at 486. We can set our formula equal to 486 and solve for \( n \): \[ 7n - 11 = 486 \] \[ 7n = 497 \] \[ n = \frac{497}{7} = 71 \] Thus, there are 71 terms in total. Finally, we can express the sum in sigma notation: \[ \sum_{n=1}^{71} (7n - 11) \] This compactly represents the series \( -4 + 3 + 10 + \ldots + 486 \) in sigma notation!