Progressions: (lesson 1 of 2)

## Arithmetic Progressions

Definition:

By an arithmetic progression of m terms, we mean a finite sequence of the form

a, a + d, a + 2d, a + 3d, . . . , a + ( m - 1)d.

The real number a is called the first term of the arithmetic
progression, and the real number d is called the difference of the arithmetic progression.

Example 1:

Consider the sequence of numbers

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

This sequence has the property that the difference between successive terms is constant and
equal to 2.

Here we have: **a = 1; d = 2.**

Example 2:

Consider the sequence of numbers

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32.

This sequence has the property that the difference between successive terms is constant and
equal to 3.

Here we have: ** a = 2; d = 3.**

### General term of arithmetic progression:

The general term of an arithmetic progression with first term a_{1} and
common difference d is:

Example 3: Find the general term for the arithmetic sequence
** -1, 3, 7, 11, . . . ** Then find **a**_{12}.

Solution:

Here **a**_{1} = 1. To find **d** subtract any two adjacent terms: ** d = 7 - 3 = 4.** The general term is:

To find **a**_{12} let **k = 12.**

Example 4: If a_{3} = 8 and a_{6} = 17, find
a_{14}.

Solution:

Use the formula for **a**_{k} with the given terms

This gives a system of two equations with two variables. We find: **a**_{1} = 2 and **d=3**.

Use the formula for **a**_{k} to find **a**_{14}

Exercise:

### Sum of an arithmetic progression:

The sum of the **n** terms of an arithmetic progression with first term a_{1} and common difference d is:

Also, the sum of an arithmetic progression is equal to

Example 5: Find the sum of the **10** terms of the arithmetic progresion
if **a**_{1} = 5 and **d = 4**.

Solution:

Example 6: Find 1 + 2 + 3 + . . . + 100

Solution:

In this example we have: **a**_{1} = 1, d = 1, n = 100, a_{100} = 100. The sum is:

Exercise: