Answer
$10,036$
Work Step by Step
We have to determine the sum:
$S=4+4.5+5+5.5+...+100$
As $4.5-4=5-4.5=5.5-5=...=0.5$, the sequence is arithmetic.
Determine its first term and the common difference:
$a_1=4$
$d=0.5$
Determine the number of terms:
$a_n=a_1+(n-1)d$
$a_n-a_1=(n-1)d$
$n-1=\dfrac{a_n-a_1}{d}$
$n=\dfrac{a_n-a_1}{d}+1$
$n=\dfrac{100-4}{0.5}+1$
$n=193$
Therefore, the given sum contains $193$ terms, so we have to determine the sum of the first $193$ terms. We use the formula:
$S_n=\dfrac{n(a_1+a_n)}{2}$
$4+4.5+5+5.5+...+100=\dfrac{193(100+4)}{2}$
$=\dfrac{193\cdot 104}{2}$
$=10,036$
