How To Find The Sum Of Arithmetic Sequences With Examples

In mathematics, arithmetic sequences are a fundamental concept. An arithmetic sequence (or arithmetic progression) is a sequence of numbers such that the difference between the consecutive terms is constant. Finding the sum of an arithmetic sequence is a common problem in algebra and calculus. This article provides a detailed explanation of how to calculate the sum of various arithmetic sequences, complete with examples.

Understanding Arithmetic Sequences

Before diving into calculating sums, let's clarify what an arithmetic sequence is. In arithmetic sequences, the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.

Key Components of an Arithmetic Sequence

To effectively work with arithmetic sequences, understanding the key components is crucial:

• First Term (a₁): This is the first number in the sequence.
• Common Difference (d): The constant difference between consecutive terms.
• Number of Terms (n): The total number of terms in the sequence.
• nth Term (aₙ): The term at position 'n' in the sequence.

Formula for the nth Term

The formula to find the nth term (aₙ) of an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

This formula helps in identifying any term in the sequence without having to list all the terms before it.

The Sum of an Arithmetic Sequence

The sum of the first 'n' terms of an arithmetic sequence, denoted as Sₙ, can be calculated using the following formula:

Sₙ = n/2 * (a₁ + aₙ)

Where:

• Sₙ is the sum of the first n terms.
• n is the number of terms.
• a₁ is the first term.
• aₙ is the nth term.

Alternatively, if you don't know the last term (aₙ), you can use the following formula:

Sₙ = n/2 * [2a₁ + (n - 1)d]

This formula only requires the first term, the common difference, and the number of terms.

Example Problems

Let's apply these formulas to several examples to illustrate how to find the sum of arithmetic sequences.

1) Find the sum of the arithmetic sequence: 3, 10, 17, 24, 31, 38, 45

In this sequence:

• a₁ (first term) = 3
• d (common difference) = 10 - 3 = 7
• n (number of terms) = 7
• a₇ (last term) = 45

Using the formula Sₙ = n/2 * (a₁ + aₙ):

S₇ = 7/2 * (3 + 45)
S₇ = 7/2 * 48
S₇ = 7 * 24
S₇ = 168

Therefore, the sum of the arithmetic sequence 3, 10, 17, 24, 31, 38, 45 is 168.

2) Find the sum of the arithmetic sequence: 9, 5, 1, -3, -7

In this sequence:

• a₁ = 9
• d = 5 - 9 = -4
• n = 5
• a₅ = -7

Using the formula Sₙ = n/2 * (a₁ + aₙ):

S₅ = 5/2 * (9 + (-7))
S₅ = 5/2 * 2
S₅ = 5

The sum of the arithmetic sequence 9, 5, 1, -3, -7 is 5.

3) Find the sum of the arithmetic sequence: 102, 96, 90, 84

In this sequence:

• a₁ = 102
• d = 96 - 102 = -6
• n = 4
• a₄ = 84

Using the formula Sₙ = n/2 * (a₁ + aₙ):

S₄ = 4/2 * (102 + 84)
S₄ = 2 * 186
S₄ = 372

The sum of the arithmetic sequence 102, 96, 90, 84 is 372.

4) Find the sum of the arithmetic sequence: 1, 15, 29, 43, 57

For the sequence 1, 15, 29, 43, 57:

• a₁ = 1
• d = 15 - 1 = 14
• n = 5
• a₅ = 57

Using the formula Sₙ = n/2 * (a₁ + aₙ):

S₅ = 5/2 * (1 + 57)
S₅ = 5/2 * 58
S₅ = 5 * 29
S₅ = 145

Thus, the sum of the arithmetic sequence 1, 15, 29, 43, 57 is 145.

5) Find the sum of the first 10 terms of the arithmetic sequence: 4, 11, 18, 25, ...

Here, we need to find the sum of the first 10 terms. We have:

• a₁ = 4
• d = 11 - 4 = 7
• n = 10

We don't have a₁₀, so we first find it using the formula aₙ = a₁ + (n - 1)d:

a₁₀ = 4 + (10 - 1) * 7
a₁₀ = 4 + 9 * 7
a₁₀ = 4 + 63
a₁₀ = 67

Now, we use the sum formula Sₙ = n/2 * (a₁ + aₙ):

S₁₀ = 10/2 * (4 + 67)
S₁₀ = 5 * 71
S₁₀ = 355

The sum of the first 10 terms of the arithmetic sequence is 355.

6) Find the sum of the first 12 terms of the arithmetic sequence: 1, 1 1/2, 2, ...

First, let's rewrite the sequence as 1, 1.5, 2, ...

• a₁ = 1
• d = 1.5 - 1 = 0.5
• n = 12

We need to find the 12th term:

a₁₂ = 1 + (12 - 1) * 0.5
a₁₂ = 1 + 11 * 0.5
a₁₂ = 1 + 5.5
a₁₂ = 6.5

Now, we use the sum formula:

S₁₂ = 12/2 * (1 + 6.5)
S₁₂ = 6 * 7.5
S₁₂ = 45

Therefore, the sum of the first 12 terms is 45.

7) Find the sum of the first 6 terms of the arithmetic sequence: -15, -10, -5, ...

In this sequence:

• a₁ = -15
• d = -10 - (-15) = 5
• n = 6

First, we find the 6th term:

a₆ = -15 + (6 - 1) * 5
a₆ = -15 + 5 * 5
a₆ = -15 + 25
a₆ = 10

Now, we calculate the sum:

S₆ = 6/2 * (-15 + 10)
S₆ = 3 * (-5)
S₆ = -15

The sum of the first 6 terms is -15.

Conclusion

Finding the sum of arithmetic sequences involves understanding the sequence's properties and applying the appropriate formula. Whether you know the last term or not, the formulas Sₙ = n/2 * (a₁ + aₙ) and Sₙ = n/2 * [2a₁ + (n - 1)d] provide a straightforward way to calculate the sum. By mastering these concepts and formulas, you can efficiently solve a wide range of problems involving arithmetic sequences. These arithmetic sequences and series are not just mathematical exercises; they form the bedrock for various real-world applications, such as financial calculations, physics problems, and computer algorithms. The ability to work with arithmetic progressions is, therefore, a valuable skill in many fields.

By practicing with different examples and variations, you can enhance your understanding and proficiency in dealing with arithmetic sequences and their sums. The examples provided in this guide offer a solid foundation for tackling more complex problems and appreciating the elegance of mathematical patterns.