What is the sum of all positive integers smaller than $1000$ that can be written in the form $100\cdot 2^n$, where $n$ is an integer (not necessarily positive)

The sum is 1575.

Step-by-step explanation:

Consider the provided information.

It is given that positive integers smaller than 1000 and that can be written in the form

Where n is integer that means the value of n can be a positive number or a negative number.

For n = 0

For n=-1

For n=-2

For n = -3 the obtained number is not an integer.

Now consider the positive value of n.

For n=1

For n=2

 For n=3

For n=4 the obtained number is greater than 1000.

Now add all the numbers.

Hence, the sum is 1575.

111

step-by-step explanation: