Answer
The sequence is arithmetic with $d=\dfrac{-3}{4}$.
and
$S_{50} =-\dfrac{2225}{4}$
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 that: $r=\dfrac{a_n}{a_{n−1}}$
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:
$a_1=8-(\dfrac{3}{4})(1)=\dfrac{29}{4}
\\a_2=8-(\dfrac{3}{4})(2)=\dfrac{13}{2}
\\a_3=8-(\dfrac{3}{4})(3)=\dfrac{23}{4}$
We see that the values decrease by $\dfrac{3}{4}$, so the sequence is arithmetic with
$d=a_n-a_{n-1}=a_2-a_1=\dfrac{13}{2}-\dfrac{29}{4}=\dfrac{-3}{4}$.
In order to find the sum of the first 50 terms, we will substitute $a_1=\dfrac{29}{4}$ and $d=\dfrac{-3}{4}$ into the formula in (1) to obtain:
$S_{50}=\dfrac{50}{2} [(2)(\dfrac{29}{4})+(\dfrac{-3}{4})(50−1)]=25(-\dfrac{89}{4})=-\dfrac{2225}{4}$
Therefore, the sum of first 50 terms is equal to: $S_{50} =-\dfrac{2225}{4}$
