A series where each term (except the first) exceeds the previous term by a constant is called an Arithmetic Series or Arithmetic Progression which we will shorten to A.P. The constant is called the common difference $d$. Any A.P. starts with some number $a_1$ which is the first term of the series. Thus the A.P. with first term $a_1=5$, common difference $d=3$ and a total of $n=6$ terms is:

$$5+8+11+14+17+20$$

Consider the numbers 0 and 30. Their arithmetic mean is 15 because 0, 15, 30 are in arithmetic progression with $a_1 =0$ and $d = 15$. However 0, 10, 20, 30 are also in arithmetic progression this time with $a_1 = 0$ and $d = 10$. So 10 and 20 are two arithmetic means between 0 and 30. Similarly 7.5, 15 and 22.5 are three arithmetic means between 0 and 30. In general we may insert any number of arithmetic means between any two numbers.

Exercises

1. Show that the $n$ th term of an A.P. is given by

$$a_n=a_1+(n\mathbin{-}1)d$$

2. Show that a series whose $n$ th term is linear in $n$ and given by $a_n = pn + q$ is an A.P. Find its first term and common difference in terms of the constants $p$ and $q$.

3.  Show that the arithmetic mean of $a$ and $b$ is the average $\frac{a + b}{2}$ of $a$ and $b$

4. Insert five arithmetic means between 19 and 37.

5. Find the number of positive multiples of 7 less than 1000.

6. The $m$ th term of an A.P. is $m^2$ and the $n$ th term is $n^2$. Find the first term and common difference.

7. The $m$ th term of an A.P. is $M$ and the $n$ th term is $N$. Find the $(m + n)$ th term and the $(m\mathbin{-}n)$ th term in terms of $M$ and $N$.

8. The $(m+n)$ th term of an A.P. is $x$ and the $(m\mathbin{-}n)$ th term is $y$. Find the $m$ th and $n$ th terms in terms of $x$ and $y$.