A series where each term exceeds the previous one by a constant is called an Arithmetic Series or Arithmetic Progression which we will abbreviate as A.P. An A.P. starts with some number $a_1$ which is the first term of the series. The constant by which any term exceeds the previous term is called the common difference and is represented by the letter $d$. Thus, the A.P. with first term $a_1 =5$, common difference $d = 3$ and a total of eleven terms is:

$$ 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35$$

Consider the first and last terms i.e. 5 and 35. The arithmetic mean of 5 and 35 is 20 because 5, 20 and 35 form an A.P. with $a_1 =5$ and $d = 15$. However, we don’t just have a single arithmetic mean. The numbers 5, 15, 25, 35 also form an A.P. this time with $a_1 = 5$ and $d = 10$. So we have inserted two arithmetic means (15 and 25) between 5 and 35. 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 $y$.