Question

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)

Answer 1

The sum is 1575.

Explanation:

Consider the provided information.

It is given that positive integers smaller than 1000 and that can be written in the form
`100\cdot 2^n`

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

For n = 0

`100\cdot 2^(0)=100`

For n=-1

`100\cdot 2^(-1)=50`

For n=-2

`100\cdot 2^(-2)=25`

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

Now consider the positive value of n.

For n=1

`100\cdot2^1 = 200`

For n=2

`100\cdot2^2 = 400`

For n=3

`100\cdot2^3 = 800`

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

Now add all the numbers.

`100+50+25+200+400+800=1575`

Hence, the sum is 1575.