Arithmetic Sequences

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by d. The general form of an arithmetic sequence can be expressed as:

a_n = a₁ + (n - 1)d

where:

• a_n is the nth term of the sequence,
• a₁ is the first term,
• d is the common difference, and
• n is the term number.

Finding the Common Difference

To find the common difference of an arithmetic sequence, subtract the first term from the second term. For example, consider the sequence 3, 7, 11, 15:

Common difference (d) = a2 - a1 = 7 - 3 = 4

Thus, the common difference is 4.

Finding Terms of an Arithmetic Sequence

To find any term in an arithmetic sequence, use the formula mentioned earlier. For example, to find the 10th term of the sequence 3, 7, 11, 15:

• First, identify the first term a₁ = 3 and the common difference d = 4.
• Now, substitute into the formula:

a10 = a1 + (10 - 1)d

a10 = 3 + (9)(4)

a10 = 3 + 36 = 39

The 10th term of the sequence is 39.

Sum of an Arithmetic Sequence

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

S_n = n/2 × (2a₁ + (n - 1)d)

Alternatively, it can also be expressed as:

S_n = n/2 × (a₁ + a_n)

where:

• S_n is the sum of the first n terms,
• a_n is the nth term, and
• n is the number of terms.

Example: Calculating the Sum

Consider the arithmetic sequence 2, 5, 8, 11, 14. To find the sum of the first 5 terms:

• First, identify a₁ = 2 and d = 3.
• Calculate a₅:

a5 = a1 + (5 - 1)d

a5 = 2 + (4)(3)

a5 = 2 + 12 = 14

Now, substitute into the sum formula:

S5 = 5/2 × (2 × 2 + (5 - 1) × 3)

S5 = 5/2 × (4 + 12)

S5 = 5/2 × 16 = 5 × 8 = 40

The sum of the first 5 terms is 40.

Applications of Arithmetic Sequences

Arithmetic sequences can be found in various real-life situations. For example, if you save R100 every month, the amount saved forms an arithmetic sequence with a first term of R100 and a common difference of R100. The total savings after n months can be calculated using the sum formula.

Example: Savings Calculation

If you save R100 each month for 12 months, the total savings can be calculated as:

S12 = 12/2 × (2 × 100 + (12 - 1) × 100)

S12 = 6 × (200 + 1100)

S12 = 6 × 1300 = 7800

Your total savings after 12 months will be R7,800.

Recursive Definition of Arithmetic Sequences

Arithmetic sequences can also be defined recursively. This means that each term is defined in relation to the previous term. The recursive definition is:

a_n = a_n-1 + d for n > 1, with a₁ = a₁.

For example, if a₁ = 5 and d = 3, the sequence can be defined as:

a2 = a1 + d = 5 + 3 = 8

a3 = a2 + d = 8 + 3 = 11

This generates the sequence 5, 8, 11, 14, …

Check for Understanding

It is important to practice identifying and working with arithmetic sequences. Ensure you understand how to find the common difference, specific terms, and the sum of terms. Use the formulas provided to assist with your calculations.

Summary

• An arithmetic sequence has a constant difference between terms.
• The nth term can be calculated using a_n = a₁ + (n - 1)d.
• The sum of the first n terms is calculated using S_n = n/2 × (2a₁ + (n - 1)d).
• Arithmetic sequences can be defined recursively.

Check your understanding

1. What is the common difference in the sequence 10, 15, 20, 25?
2. Calculate the 7th term of the arithmetic sequence where the first term is 4 and the common difference is 3.
3. Find the sum of the first 8 terms of the arithmetic sequence 1, 4, 7, 10.
4. Define an arithmetic sequence recursively and provide an example.