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.

To find the number of terms in the series, we recognize that this is an arithmetic series (sequence), with the first term \( a = -7 \) and a common difference \( d = 4 \) (since \(-3 - (-7) = 4\)). To find the number of terms \( n \) that leads to the last term \( 173 \), we use the formula for the \( n \)-th term of an arithmetic sequence: \[ a_n = a + (n - 1)d \] Setting \( a_n = 173 \): \[ 173 = -7 + (n - 1) \cdot 4 \] Solving for \( n \): \[ 173 + 7 = (n - 1) \cdot 4 \] \[ 180 = (n - 1) \cdot 4 \] \[ n - 1 = \frac{180}{4} = 45 \] \[ n = 45 + 1 = 46 \] So, there are **46 terms** in the series. Next, to calculate the sum \( S_n \) of the series, we use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a + a_n) \] Substituting \( n = 46 \), \( a = -7 \), and \( a_n = 173 \): \[ S_{46} = \frac{46}{2} (-7 + 173) \] \[ S_{46} = 23 \cdot 166 = 3818 \] Thus, the **sum of the series is 3818**. Finally, the series in sigma notation can be expressed as: \[ \sum_{k=0}^{45} (-7 + 4k) \]