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

To express the series \( -4 + 3 + 10 + \ldots + 486 \) in sigma notation, let's first identify the pattern of the sequence. ### Step 1: Determine the Sequence Pattern The given sequence is: \[ -4, \, 3, \, 10, \, \ldots, \, 486 \] Calculate the difference between consecutive terms: \[ 3 - (-4) = 7 \\ 10 - 3 = 7 \] This indicates that the sequence is **arithmetic** with a common difference \( d = 7 \). ### Step 2: Find the General Term The general term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n - 1)d \] where: - \( a_1 = -4 \) (the first term) - \( d = 7 \) (common difference) So, \[ a_n = -4 + (n - 1) \times 7 = 7n - 11 \] ### Step 3: Determine the Number of Terms To find the number of terms (\( n \)) when the last term is 486: \[ 7n - 11 = 486 \\ 7n = 497 \\ n = 71 \] ### Step 4: Write in Sigma Notation Putting it all together, the series can be expressed in sigma notation as: \[ \sum_{k=1}^{71} (7k - 11) \] Alternatively, you can adjust the index to start from 0: \[ \sum_{k=0}^{70} (7k - 4) \] Both representations are correct and represent the same series. **Final Answer:** \[ \sum_{k=0}^{70} \left(7\,k\;-\;4\,\right) \] **Or equivalently:** \[ \sum_{k=1}^{71} \left(7k - 11\right) \]