Arithmetic number sequence

2020-12-01T17:55:08+02:00

A number sequence is increasing, if each term in it is either greater than or equal to the previous term. In contrast, the sequence is decreasing, if each term in it is less than or equal to the previous term.

The set of natural numbers {[[$ 0, 1, 2, 3,… $]]} is an example of an increasing number sequence. In it, each term is obtained by adding the number [[$ 1 $]] to the previous one. Such a sequencen in which the term is obtained from the previous by adding the same constant to itn is called an arithmetic number sequence.

Let's examine how the general term [[$ a_n $]] of an arithmetic sequence is determined by the first term of the sequence [[$ a_1 $]] and the constant [[$ d $]].

[[$ \quad \begin{align*} a_2 &= a_1 + d \ \\ a_3 &= a_2 + d = a_1 + d + d = a_1 + 2d \ \\ a_4 &= a_3 + d = a_1 + 2d + d = a_1 + 3d \ \\ a_n &= a_1 + (n - 1)d \end{align*} $]]

In an arithmetic number sequence, the difference between two consecutive terms is constant.

The general term for an arithmetic sequence is [[$ a_n = a_1 + (n - 1) d $]], where [[$ a_1 $]] is the first term in the number sequence and [[$ d $]] is the difference number.

Example 1

Natural numbers divisible by [[$ 10 $]] form an arithmetic sequence {[[$ 0, 10, 20, 30,… $]]}. The difference number [[$ d $]] of the sequence is [[$ 10 $]], and the general term of the sequence is

[[$ \begin{align*} a_n &= 0 + (n -1) \cdot 10 \\ &= 10n - 10\end{align*} $]]

Example 2

Oscar starts saving for a tablet computer. He already has [[$ 100 $]] QAR in savings, and every week he puts additional [[$ 8 $]] QAR in his savings. Calculate how much money Oscar has after [[$ 20 $]] weeks.

This is an arithmetic number sequence, the first term of which is [[$ a_1 = 100 $]] and the difference number of which is [[$ d = 8 $]]. These give a general term for the sequence as follows:

[[$ \begin{align*} a_n &= 100 + (n -1) \cdot 8 \\ &= 100 + 8n - 8\\ & = 92 + 8n,\end{align*} $]]

This can be used to calculate the [[$ 20 $]]th term of the sequence.[[$ a_{20} = 92 + 8 \cdot 20 = 252 $]].

Answer: After [[$ 20 $]] weeks, he has saved a total of [[$ 252 $]] QAR.

9. Arithmetic number sequences

Arithmetic number sequence

Example 1

Example 2

Exercises