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Arithmetic Progressions

Arithmetic Progressions

An Arithmetic Sequence increases, or decreases, by a set amount each time. This change is known as the common difference: the difference between two terms is the same each time. The term-to-term rule is adding, or subtracting, the difference.

For example, the sequence 5, 8, 11, 14, 17, ... has a common difference between each pair of terms. That difference is 3. The next term after 17 is therefore 17 + 3 = 20.

Arithmetic sequences can also fall in value: the common difference in this case will be negative.

Arithmetic sequences can start on any number, with the common difference being applied from the first number on. In the example above, the first term was 5; with the following terms incremented by 3, the common difference.

The nth term in an arithmetic expression can be found from the formula `u_n = a + (1-n)d`, where `u_n` is the value of the nth term, `a` is the starting value of the sequence, and `d` is the common difference.

Example 1

What is first term from this sequence: ..., 9, 13, 17, 21, 25 ?

The common difference is 4; the first number in the sequence is 9 - 4 = 5.

Because it is falling, the answer is negative.

Answer: 5

Example 2

What is the common difference for this sequence: 55, 50, 45, 40, 35, ...

The difference is falling by 5 every time.

Answer: -5