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# Arithmetic Progressions

What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.

If two numbers have arithmetic mean 2700 and harmonic mean 75, then find their geometric mean.

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**Note**:

- The arithmetic mean of two numbers \(a\) and \(b\) is \(\frac{a+b}2\).
- The harmonic mean of two numbers \(a\) and \(b\) is \( \frac2{\frac1{a} + \frac1{b}} \).

\[1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \ldots + 301 + 302 = ?\]

**Clarification**: The sum keeps alternating between two distinct positive numbers and two distinct negative numbers.

Real numbers \(a_1,a_2,\ldots,a_{99}\) form an arithmetic progression.

Suppose that \[ a_2+a_5+a_8+\cdots+a_{98}=205.\] Find the value of \( \displaystyle \sum_{k=1}^{99} a_k\).

\(a_1,a_2,a_3,\ldots, a_{98} \) are terms in an arithmetic progression with common difference 1 such that their sum is 137.

What is the sum of the even terms of this progression? That is, what is the value of \( a_2+a_4+a_6+\cdots+a_{98} \)?

\[\large 54+51+48+45+ \cdots\]

You are given the sum of an arithmetic progression of a finite number of terms as shown above.

What is the minimum number of terms used to make a total value of 513?

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