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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.

# Arithmetic Progressions: Level 2 Challenges

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


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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