Arithmetic Progressions
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 \( \frac2{\frac1{a} + \frac1{b}} \).
by
Calvin Lin
\[1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + \ldots + 301 + 302 = ?\]
Clarification: The sum keeps alternating between two distinct positive numbers and two distinct negative numbers.
by
Roberto Lassari
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 \( \displaystyle \sum_{k=1}^{99} a_k\).
by
Russelle Guadalupe
\(a_1,a_2,a_3,\ldots, a_{98} \) 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 \( a_2+a_4+a_6+\cdots+a_{98} \)?
by
Sandeep Bhardwaj
\[ 54+51+48+45+ \cdots\]
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?
by
Sravanth Chebrolu