QUESTION 6 Given the arithmetic series: \( -7-3+1+\cdots+173 \) 6.1 How many terms are there in the series? 6.2 Calculate the sum of the series. Write the series in sigma notation.

1. **Identify the first term and common difference** The first term is \[ a_1 = -7 \] The second term is \(-3\), so the common difference is \[ d = -3 - (-7) = 4. \] 2. **Determine the number of terms** The \(n\)th term of an arithmetic series is given by \[ a_n = a_1 + (n-1)d. \] Given that the last term \(a_n = 173\), substitute the known values: \[ 173 = -7 + (n-1) \cdot 4. \] Solve for \(n\): \[ 173 + 7 = 4(n-1) \] \[ 180 = 4(n-1) \] \[ n-1 = \frac{180}{4} = 45 \] \[ n = 45 + 1 = 46. \] Thus, there are \(46\) terms in the series. 3. **Calculate the sum of the series** The sum \(S_n\) of an arithmetic series is given by \[ S_n = \frac{n}{2}(a_1 + a_n). \] Substitute the values: \[ S_{46} = \frac{46}{2}(-7 + 173). \] Calculate inside the parentheses: \[ -7 + 173 = 166. \] Now compute the sum: \[ S_{46} = 23 \times 166 = 3818. \] Therefore, the sum of the series is \(3818\). 4. **Write the series in sigma notation** One way to express the series in sigma notation is: \[ \sum_{k=1}^{46} \left(-7 + 4(k-1)\right). \]