Arithmetic Progressions

Arithmetic Progressions: Level 2 Challenges


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


  • The arithmetic mean of two numbers aa and bb is a+b2\frac{a+b}2.
  • The harmonic mean of two numbers aa and bb is 21a+1b \frac2{\frac1{a} + \frac1{b}} .

1+234+5+678++301+302=?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 a1,a2,,a99a_1,a_2,\ldots,a_{99} form an arithmetic progression.

Suppose that a2+a5+a8++a98=205. a_2+a_5+a_8+\cdots+a_{98}=205. Find the value of k=199ak \displaystyle \sum_{k=1}^{99} a_k.

a1,a2,a3,,a98a_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 a2+a4+a6++a98 a_2+a_4+a_6+\cdots+a_{98} ?

54+51+48+45+ 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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