Title: 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 finite 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 finite 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.

k-th term of arithmetic progression:

The k  th term of an arithmetic progression is equal to

Sum of an arithmetic progression:

The sum of the m terms of an arithmetic progression of the type is equal to

Note that the sum of an arithmetic progression is equal to

Example 3:

Suppose that the 4-th and 7-th terms of an arithmetic progression are equal to 9 and -15 respectively. Then we have

9 = a + 3d

-15 = a + 6d

so that 3d = 24. It follows that d = 8 and a = 33. The arithmetic progression is given by

33, 25, 17, 9, 1, -7, -15, . . .

The 10-th term is given by a + 9d = 33 -9*8 = -39. The sum of the first 10 terms is equal to

Important formulas

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