Answer
The sequence is arithmetic.
common difference = $d=1$
Sum of first 50 terms = $S_{50}=1375$
Work Step by Step
RECALL:
(1) A sequence is arithmetic if there exists a common difference $d$ among consecutive terms.
$d=a_n−a_{n−1}$
The sum of the first n terms of an arithmetic sequence is given by the formulas:
$S_n=\frac{n}{2}(a_1+a_n)$ or $S_n=\frac{n}{2}[2a_1+(n−1)d]$
(2) A sequence is geometric if there exists a common ratio $r$ among consecutive terms.
$r=\frac{a_n}{a_{n−1}}$
The sum of the first n terms of a geometric sequence is given by the formula:
$S_n=a_1\cdot\dfrac{1−r^n}{1−r}$
In the formulas listed above,
d = common difference
r = common ratio
$a_1$ = first term
$a_n$ = nth term
n = number of terms
List the first few terms of the sequence.
Substitute 1,2,3 to n to list the first three terms:
$a_1=1+2=3
\\a_2=2+2=4
\\a_3=3+2=5$
Since the values increase by 1, the sequence is arithmetic with $d=1$.
Find the sum of the first 50 terms:
With $a_1=3$ and $d=1$, solve for the sum of the first 50 terms using the formula in (1) above to obtain:
$Sn=\frac{n}{2}(2a_1+(n−1)d))
\\S_{50}=\frac{50}{2}(2⋅3+1(50−1))
\\S_{50}=25(6+1⋅49)
\\S_{50}=25(6+49)
\\S_{50}=25(55)
\\S_{50}=1375$
