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Sequences and Series

What's the sum of the first 100 positive integers? How about the first 1000?

Level 1

         

How many squares are present in figure number 20?

When I was a teenager, a friend and I stole soda from our parents. I took the soda bottle, drank half of what's there, and then passed it to my friend. My friend drank half of what was left, and passed it back to me. I again drank half of what was left, and passed it back to him.
We kept up with this until the entire soda bottle was empty.

Approximately what percentage of the soda did I drink?

\[\large \frac{100001+100003+100005+\cdots+199999}{1+3+5+7+\cdots+99999} = \ ? \]

\[\left(\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\ldots\right) + \\ \hspace{1cm}\left(\dfrac1{2^3}+\dfrac1{3^3}+\dfrac1{4^3} + \ldots\right) + \\ \hspace{2.5cm}\left(\dfrac1{2^4}+\dfrac1{3^4}+\dfrac1{4^4}+\ldots\right)+\cdots\]

What is the value of the series above?

A person puts \(1\) grain of rice in the first square of a chess board. In the subsequent square she puts twice that of the previous square. How many total grains does she need to fill out all of the squares of the chess board?

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