Arithmetic sequences and series

Arithmetic Sequences and Series

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by d. For example, in the sequence 2, 5, 8, 11, the common difference is 3, since each term increases by 3.

Defining an Arithmetic Sequence

The nth term of an arithmetic sequence can be expressed using the formula:

a_n = a_1 + (n - 1)d

Where:

• a_n = the n-th term of the sequence
• a_1 = the first term of the sequence
• d = the common difference
• n = the term number

Example 1: Finding the n-th Term

Consider the arithmetic sequence: 4, 9, 14, 19, ...

Here, a_1 = 4 and the common difference d = 9 - 4 = 5.

To find the 10th term (a_{10}), substitute the values into the formula:

a_{10} = a_1 + (10 - 1)d

a_{10} = 4 + (9)(5)

a_{10} = 4 + 45 = 49

Thus, the 10th term is 49.

Defining an Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms of an arithmetic series can be calculated using the formula:

S_n = rac{n}{2} (a_1 + a_n)

Alternatively, if you know the common difference, you can use:

S_n = rac{n}{2} [2a_1 + (n - 1)d]

Where S_n is the sum of the first n terms.

Example 2: Finding the Sum of the First n Terms

Using the previous sequence (4, 9, 14, 19, ...), let’s find the sum of the first 10 terms.

First, we already know:

• a_1 = 4
• a_{10} = 49
• n = 10

Using the sum formula:

S_{10} = rac{10}{2} (4 + 49)

S_{10} = 5 imes 53 = 265

The sum of the first 10 terms is 265.

Common Mistakes

Identifying Arithmetic Sequences

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. For example, consider the sequence 7, 10, 13, 16:

• 10 - 7 = 3
• 13 - 10 = 3
• 16 - 13 = 3

Since the difference is constant, this is an arithmetic sequence with d = 3.

Example 3: Identifying an Arithmetic Sequence

Examine the sequence: 2, 4, 8, 10. Calculate the differences:

• 4 - 2 = 2
• 8 - 4 = 4
• 10 - 8 = 2

Since the differences are not constant, this sequence is not arithmetic.

Applications of Arithmetic Sequences

Arithmetic sequences have real-world applications. For example, if you save R100 in the first month and increase your savings by R50 each month, your savings form an arithmetic sequence.

Example 4: Real-World Application

If you save R100 in the first month, the second month you save R150, the third month R200, and so on. This sequence is 100, 150, 200, ...

Here, a_1 = 100 and d = 50. To find how much you will save in the 12th month:

a_{12} = a_1 + (12 - 1)d

a_{12} = 100 + (11)(50)

a_{12} = 100 + 550 = 650

In the 12th month, you will save R650.

Summary

• An arithmetic sequence has a constant difference between terms.
• The n-th term can be calculated using a_n = a_1 + (n - 1)d.
• The sum of the first n terms can be found using S_n = rac{n}{2} (a_1 + a_n) or S_n = rac{n}{2} [2a_1 + (n - 1)d].
• Check for a constant difference to identify an arithmetic sequence.

Check your understanding

1. Find the 15th term of the arithmetic sequence: 3, 7, 11, 15, ...
2. Calculate the sum of the first 20 terms of the arithmetic sequence: 5, 10, 15, ...
3. Determine if the sequence 10, 20, 30, 45 is an arithmetic sequence.
4. If you start with R200 and save R100 more each month, how much will you have saved in the 6th month?