Algebra
# 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?

\[ \large p^\text{th} \ \text{term } + q^\text{th} \ \text{term } + r^\text{th} \ \text{term } \]

\[\dfrac{a(q-r)}{p} + \dfrac{b(r-p)}{q} + \dfrac{c(p-q)}{r}\]

In an arithmetic progression, the sum of the first \(p, q, r\) terms are \(a, b, c\) respectively. Compute the expression above.

\[ \begin{array}{llllll} A_1 : & 2, & 9, & 16, \ldots , & 2 + (1000-1) \times 7 \\ A_2: & 3, & 12 , & 21, \ldots, & 3 + (1000-1) \times 9? \\ \end{array} \]

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.

\[\large \frac ab \ , \ ab \ , \ a -b \ , \ a+b \]

Above shows real numbers that belong to an arithmetic progression in order. Find the next term of this sequence.