Answer
The sequence is arithmetic with $d=\dfrac{-2}{3}$.
and
$S_{50} =-700$
Work Step by Step
We will check whether the sequence is arithmetic or geometric.
(1) A sequence is arithmetic if there exists a common difference $d$ among consecutive terms such as: $d=a_n−a_{n−1}$
The sum of the first n terms of an arithmetic sequence can be computed as:
$S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$
(2) A sequence is geometric if there exists a common ratio $r$ among consecutive terms such as: $r=\dfrac{a_n}{a_{n−1}}$
The sum of the first $n$ terms of a geometric sequence can be computed as:
$S_n= \dfrac{a_1(1-r^n)}{1-r} (2)$
where, $a_1 =\ First \ term$, $a_n$ = $n$th term, and $n =\ Number \ of \ Terms$
Substitute $n=1,2,3$ to list the first three terms:
$a_1=3-(\dfrac{2}{3})(1)=\dfrac{7}{3}
\\a_2=3-(\dfrac{2}{3})(2)=\dfrac{5}{3}
\\a_3=3-(\dfrac{2}{3})(3)=\dfrac{3}{3}=1$
We see that the values decrease by $\dfrac{2}{3}$. Thus, the sequence is arithmetic with
$d=a_n-a_{n-1}=a_2-a_1=\dfrac{5}{3}-\dfrac{7}{3}=\dfrac{-2}{3}$.
In order to find the sum of the first 50 terms, we will substitute $a_1=\dfrac{7}{3}$ and $d=\dfrac{-2}{3}$ into the formula in (1) to obtain:
$S_{50}=\dfrac{50}{2} [(2)(\dfrac{7}{3})+(\dfrac{-2}{3})(50−1)]=25(-\dfrac{84}{3})=-700$
Therefore, the sum of first 50 terms is equal to: $S_{50} =-700$
