My favorite 3 concepts in 5 problems!

Number Theory Level 3

Here it is again! You know the drill! If not, try out the Super Bowl Question.

Or hear the quote:

Directions are to solve five problems, and enter in the answer in the box like \(a_1a_2a_3a_4a_5\), where as the subscript number is the problem number. (eg. if final result is 54321, then answer to first problem is 5, second is 4, etc.) So let's begin! Note that not all answers are single digit answers.

1 [EASY]) What is le \(2000^2 - 1999^2\)? (yeah I know its on FTW)

2 [EASY]) Find the sum of the first 100 positive integers.

3 [Meh]) Find the sum of the first 200 positive integers.

4 [Heh?]) Expanding the expression \((a+b)^4\) will give you many terms, including the term \(a^3b^1\). What is the coefficient of the term?

5 [...]) Assume \(N\) is the coefficient of the term \(a^5b^2\) from the expanded expression \((a+b)^7\). Assume \(M\) is \((N)^2 - (N-1)^2\). Find the sum of the first M positive integers.


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