Arithmetic Progressions
\[ \left (1 + 3 + 5 + \ldots + p^\text{th} \right ) \\ + \\ \left (1 + 3 + 5 + \ldots + q^\text{th} \right ) \\ = \\ \left (1 + 3 + 5 + \ldots + r^\text{th} \right ) \]
Given the above equation for positive integers \(p,q,r\) with \(p^\text{th} \) term greater than 39.
What is the smallest possible value of the expression below?
\[ \large p^\text{th} \ \text{term } + q^\text{th} \ \text{term } + r^\text{th} \ \text{term } \]
by
jaiveer shekhawat
\[\dfrac{a(q-r)}{p} + \dfrac{b(r-p)}{q} + \dfrac{c(p-q)}{r}\]
In an arithmetic progression, the sum of the first \(p, q, r\) terms are \(a, b, c\) respectively. Compute the expression above.
by
Sridhar Sri
Consider an arithmetic progression with 2 and 101 as its first term and last term respectively. If the sum of the first 5 terms of this arithmetic progression is 40, find the sum of the last 5 terms of this progression.
by
Ashish Menon
\[ \begin{array}{llllll} A_1 : & 2, & 9, & 16, \ldots , & 2 + (1000-1) \times 7 \\ A_2: & 3, & 12 , & 21, \ldots, & 3 + (1000-1) \times 9? \\ \end{array} \]
How many integers appear in both of the following arithmetic progressions above?
Details and assumptions
Since 2 appears in \(A_1\) but not in \(A_2\), it does not appear in both of the arithmetic progressions.
by
Calvin Lin
\[\large \frac ab \ , \ ab \ , \ a -b \ , \ a+b \]
Above shows real numbers that belong to an arithmetic progression in order. Find the next term of this sequence.
by
Gerardo Hernandez