XIII. Arithmetical Progression

Arithmetical Progression.

A series of numbers is said to form an Arithmetical Progression if the difference between any term and the preceding term is the same throughout the series.

Thus $a$, $b$, $c$, $d$, etc., are in arithmetical progression if $b-a$, $c-b$, $d-c$, etc., are all equal.

This difference is called the common difference of the progression, and is represented by $d$. If $d$ is positive, the progression is an increasing series; if $d$ is negative, the progression is a decreasing series.

What is the common difference in each of the following series? $$\begin{gathered} r*4>r1,\&\phantom{-}4,\&7,\&10,\&\dots \\ 5,\&7,\&9,\&11,\&\dots \\ 10,\&9,\&8,\&7,\&\dots \\ 7,\&3,\&-1,\&-5,\&\dots \end{gathered}$$

If the first term of an arithmetical progression is represented by $a$ and the common difference by $d$, then {2} &\text{the \emph{second} term will be } && a + d, \ &\text{the \emph{third} term will be } && a + 2d, \ &\text{the \emph{fourth} term will be } && a + 3d,

and so on, the coefficient of $d$ in each term being always less by $1$ than the number of the term.

Hence the $n$th term will be $a+(n-1)d$.

If we represent the $n$th term by $l$, we have $l=a+(n-1)d.\text{Formula (1)}$

We can, therefore, find any term of an arithmetical progression if the first term and common difference are given, or if any two terms are given.

1. Find the 10th term of an arithmetical progression if the 1st term is $3$ and the common difference is $4$.

By formula (1), the 10th term is $3+(10-1)4$, or $39$.

2. If the 8th term of an arithmetical progression is $25$, and the 23d term $70$, find the series.

lint By formula (1), the 23d term is $a+22d$, lintand the 8th term is $a+7d$.

lint Therefore, a + 22d = 70 lintand a + 07d = 25. [1] lint Subtract, 15d = 45 lintand d = 3, lintwhence a = 4.

The series is therefore $4$, $7$, $10$, $13$, etc.

169. Arithmetical Mean.

If three numbers are in arithmetical progression, the middle number is called the arithmetical mean of the other two numbers.

If $a$, $A$, $b$ are in arithmetical progression, $A$ is the arithmetical mean of $a$ and $b$. Hence, by the definition of an arithmetical series,