Answer
Arithmetic
Sum: $-700$
Work Step by Step
We are given the sequence:
$\left\{3-\dfrac{2}{3}n\right\}$
Compute the difference between two consecutive terms:
$\left(3-\dfrac{2}{3}(n+1)\right)-\left(3-\dfrac{2}{3}n\right)=3-\dfrac{2}{3}n-\dfrac{2}{3}-3+\dfrac{2}{3}n=-\dfrac{2}{3}$
As the difference between any consecutive terms is constant, the sequence is ARITHMETIC. Determine its first element and common difference:
$a_1=3-\dfrac{2}{3}(1)=\dfrac{7}{3}$
$d=-\dfrac{2}{3}$
Compute the sum of the first 50 terms:
$S_n=\dfrac{n(2a_1+(n-1)d)}{2}$
$S_{50}=\dfrac{50\left(2\left(\dfrac{7}{3}\right)+(50-1)\left(-\dfrac{2}{3}\right)\right)}{2}=-700$
