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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)

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Answer:

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.

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