**Using arithmetic sequences**

An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant

Simply put, the next term is obtained by adding a certain number to the previous one. This number is constant, that is, it stays the same all the time.

# 2,5,8,11,14,...

Above, is an arithmetic sequence, since the difference between two consecutive terms is 3. This number is denoted by *d* also known as the common difference. The sequence would continue 17,20,23,26, ...

**Example 1**

The first term of an arithmetic sequence is 2 and the second term is 6. Work out the next two terms of the sequence.

### General term

In an arithmetic sequence we have *d* which is added to a term to get the next term.

To get the second term, we add *d* to the first term. To get the third term we add *d* again, that is, to the first term we add *d** * twice to get the third term. The fourth term is obtained by adding

*d*three times to the first term. And so on.

Therefore, we can get any term by adding *d* to the first term one time less than the order number of that term we are seeking.

**Example 2**

The first term of an arithmetic sequence is 2 and the second term is 6. Work out the tenth term of the sequence.

When finding the sum of an arithmetic series, we need the first term, the last term and a quantity of terms

**Arithmetic sequences as a model**

**Arithmetic sequences as a model**

**Example 3**

A €10 000 loan with a fixed amortisation schedule is repaid in equal payments every year. The amount for one payment is €1000. The interest on the loan is 2%, which is paid in connection to the payment. How much interest will be paid over the entire period of the loan?

The first interest is paid on the entire €10 000 of the loan capital. The loan will be reduced by €1000 and the following year the capital is €9000, from which interest will be paid. After a year €8000 etc.

**Interest**

It is seen that the amount of interest decreases steadily by €20 per year. As the capital decreases steadily by €1000 per year, the interest payable also decreases by €20 per year. The amount of interest thus forms an arithmetic sequence. There are 10 payments. The first interest to be paid is €200 and the last interest to be paid is €20, that is 2% of the remaining capital in the last year €1,000.

The interest paid for the entire loan period is €1100.

**Example 4**

At the edge of a courtyard road, 20 birch trees have been planted at regular intervals so that the fourth birch is at a distance of 35 meters from the start of the road and the tenth birch is at a distance of 65 meters from the start of the road. The length of the courtyard road is 120 meters. How far is the first birch tree from the start of the road?

Because the birches are planted at regular intervals, their distances from the start of the road form an arithmetic sequence. The fourth member is therefore 35 and the tenth member 65.

We can use a general term formula that can be generalised from two terms as follows

Thus we have

We substitute 4th and 10th terms

The distance between the birches is therefore five meters. Now we can solve the first term which is the answer to the question.

The first birch is therefore 20 meters from the start of the road.

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