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, use the information provided. The first term \( a_1 = 51 \) and the eighth term \( a_8 = 100 \). The eighth term can be expressed as: \[ a_8 = a_1 + 7d \] Substituting the known values: \[ 100 = 51 + 7d \] Solving for \( d \): \[ 49 = 7d \] \[ d = 7 \] Now, we can find the twentieth term \( a_{20} \): \[ a_{20} = a_1 + 19d \] \[ a_{20} = 51 + 19 \times 7 \] \[ a_{20} = 51 + 133 = 184 \] Next, to find the sum of the first twenty terms \( S_{20} \): \[ S_n = \frac{n}{2}(a_1 + a_n) \] Where \( n = 20 \) and \( a_{20} = 184 \): \[ S_{20} = \frac{20}{2}(51 + 184) \] \[ S_{20} = 10(235) = 2350 \] So, the answers are: a) The twentieth term of the series is \( 184 \). b) The sum of the first twenty terms of the series is \( 2350 \).