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

Answer:

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 $100\cdot 2^n$

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.