Answer
The sequence is arithmetic with $d=2$.
and
$S_{50}=2300$
Work Step by Step
A sequence is arithmetic if there exists a common difference $d$ among consecutive terms such that: $d=a_n−a_{n−1}$
The sum of the first $n$ terms of an arithmetic sequence can be computed as:
$S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$
where, $a_1 =\ First \ term$, $a_n$ = $n$th term, and $n =\ Number \ of \ Terms$
To identify the sequence as arithmetic or geometric, we substitute $n=1,2,3$ to list the first three terms to obtain:
$a_1=(2)(1)-5=-3
\\a_2=(2)(2)-5=-1
\\a_3=(2)(3)-5=1$
We see that the values increase by $1$, so the sequence is arithmetic with
$d=a_n-a_{n-1}=a_2-a_1=-1+3=2$.
In order to find the sum of the first $50$ terms, we will substitute $a_1=-3$ and $d=2$ in the formula in (1) to obtain:
$S_{50}=\dfrac{50}{2} [(2)(-3)+2(50−1)]=25(-6+98)=2300$
Therefore, the sum of first 50 terms is equal to: $S_{50}=2300$
