Definition:

Example 1:

Consider the sequence of numbers

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

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:

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:

To find a₁₂ let k = 12.

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

Solution:

Use the formula for a_k with the given terms

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

Use the formula for a_k to find a₁₄

Exercise:

Level 1

Level 2

Sum of an arithmetic progression:

The sum of the n terms of an arithmetic progression with first term a₁ 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 a₁ = 5 and d = 4.

Solution:

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

Solution:

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

Exercise:

Level 1

Level 2