Answer
The sequence is arithmetic with $d=1$
and
$S_{50}=1375$
Work Step by Step
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:
$S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$
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:
$a_1=1+2=3
\\a_2=2+2=4
\\a_3=3+2=5$
We see that the values increase by 1, so the sequence is arithmetic with
$d=a_n-a_{n-1}=a_2-a_1=4-3=1$.
In order to find the sum of the first $50$ terms, we will substitute $a_1=3$ and $d=1$ into the formula in (1) to obtain:
$S_{50}=\dfrac{50}{2} [(2)(3)+1(50−1)]=25(6+49)=1375$
Therefore, the sum of first $50$ terms is equal to: $S_{50}=1375$
