Answer
$S_{101}=20,301$
Work Step by Step
RECALL:
(1) The sum of the first $n$ terms of an arithmetic sequence is given by the formula:
$S_n=\dfrac{n}{2}(a+a_n)$
where
$a$ = first term
$d$ = common difference
$a_n$ = $n^{th}$ term
(2) The $n^{th}$ term $a_n$ of an arithmetic sequence is given by the formula:
$a_n = a + (n-1)d$
where
$a$ = first term
$d$ = common difference
The given arithmetic sequence has:
$a=1
\\a_n = 401
\\d=5-1=4$
The formula for the partial sum requires the values of $a$, $a_n$ and $n$.
However, only $a$ and $a_n$ are known at the moment.
Solve for $n$ using the formula for $a_n$ to obtain:
$a_n = a + (n-1)d
\\401 = 1+(n-1)4
\\401-1 = (n-1)(4)
\\400=(n-1)(4)
\\\dfrac{400}{4}=\dfrac{(n-1)(4)}{4}
\\100=n-1
\\100+1=n-1+1
\\101=n$
Now that it is known that $n=101$, the sum of the first 101 terms can be computed using the formula above.
$S_{101} = \dfrac{101}{2}(1+401)
\\S_{101}=\dfrac{101}{2}(402)
\\S_{101}=20,301$
