The first term of an arithmetic series is 51 and the eighth term is 100 . a) Find the twentieth term of the series. b) Determine the sum of the first twenty terms of the series.

To find the common difference \( d \) of the arithmetic series, we can use the formula for the \( n \)-th term, \( a_n = a_1 + (n - 1)d \). Given that the first term \( a_1 \) is 51 and the eighth term \( a_8 \) is 100, we have: \[ a_8 = 51 + 7d = 100 \] \[ 7d = 100 - 51 = 49 \] \[ d = \frac{49}{7} = 7 \] a) Now, to find the twentieth term \( a_{20} \): \[ a_{20} = 51 + (20 - 1) \cdot 7 = 51 + 19 \cdot 7 = 51 + 133 = 184 \] b) To determine the sum of the first twenty terms \( S_{20} \), we use the formula for the sum of the first \( n \) terms of an arithmetic series: \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \] \[ S_{20} = \frac{20}{2} \cdot (51 + 184) = 10 \cdot 235 = 2350 \] So, the twentieth term is 184, and the sum of the first twenty terms is 2350.