Answer
$S_{100}=-505$
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:
$a=-10
\\a_n = -0.1
\\d=-9.9-(-10)=-9.9+10=0.1$
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:
$\require{cancel}
a_n = a + (n-1)d
\\-0.1 = -10+(n-1)(0.1)
\\-0.1+10 = (n-1)(0.1)
\\9.9=(n-1)(0.1)
\dfrac{9.9}{0.1}=\dfrac{(n-1)(0.1)}{0.1}
\\99=n-1
\\99+1=n-1+1
\\100=n$
Now that it is known that $n=100$, the sum of the first 100 terms can be computed using the formula above.
$S_{100} = \dfrac{100}{2}[-10+(-0.1)]
\\S_{100}=50(-10.1)
\\S_{100}=-505$
