Question

What is the sum of all positive integers up to $1000$, which are divisible by $5$ and are not divisible by $2$?

• A. $10,050$
• B. $5050$
• C. $5000$
• D. $50,000$

Answer 1

The correct option is D $50,000$

The positive integers, which are divisible by $5$, are $5,10,15,\dots ,1000.$ Out of these $10,20,30,\dots ,1000$ are divisible by $2.$

Thus, we have to find the sum of the positive integers $5,15,25,....,995$

Clearly above sequence is an AP with the first term $5$ and the common difference of $10.$

If $n$ is the number of terms in this sequence then,
$\begin{array}{cc}5+10\left(n-1\right)& =995\\ 5+10n-10& =995\\ 10n& =1000\\ n& =100\end{array}$

Now by using the formula for the sum of the first $n$ terms of an AP,

$\begin{array}{cc}{S}_{100}& =\frac{100}{2}\left[2\times 5+\left(100-1\right)10\right]\\ & =50\left(10+990\right)\\ & =50\times 1000\\ & =50000\end{array}$