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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 a1 and common difference d is:

general term of arithmetic progression

Example 3: Find the general term for the arithmetic sequence -1, 3, 7, 11, . . . Then find a12.

Solution:

Here a1 = 1. To find d subtract any two adjacent terms: d = 7 - 3 = 4. The general term is:

General term solution

To find a12 let k = 12.

General term solution


Example 4: If a3 = 8 and a6 = 17, find a14.

Solution:

Use the formula for ak with the given terms

General term solution

This gives a system of two equations with two variables. We find: a1 = 2 and d=3.

Use the formula for ak to find a14

General term solution

Exercise:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

Sum of an arithmetic progression:

The sum of the n terms of an arithmetic progression with first term a1 and common difference d is:

Arithmetic Progression Sum

Also, the sum of an arithmetic progression is equal to

Arithmetic Progression Sum

Example 5: Find the sum of the 10 terms of the arithmetic progresion if a1 = 5 and d = 4.

Solution:

Arithmetic Progression Sum

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

Solution:

In this example we have: a1 = 1, d = 1, n = 100, a100 = 100. The sum is:

Arithmetic Progression Sum

Exercise:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4