Algebra

Arithmetic Progressions

Arithmetic Progressions: Level 2 Challenges

           

200 logs are stacked in the following manner:

The bottom row has 20 logs,
the row above it has 19 logs,
the row above that has 18 logs,
and so on.

How many logs are there in the top row?

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


Note:

  • 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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