How To Find First Term Of Arithmetic Sequence

How to Find the First Term of an Arithmetic Sequence

Finding the first term of an arithmetic sequence might seem like a simple task, but understanding the underlying principles and different approaches is crucial for mastering arithmetic sequences and related mathematical concepts. This comprehensive guide will equip you with the knowledge and strategies to confidently tackle various scenarios, from straightforward problems to more complex ones involving multiple unknowns.

Understanding Arithmetic Sequences

Before diving into methods for finding the first term, let's solidify our understanding of arithmetic sequences. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference, often denoted by 'd'.

For example:

• 2, 5, 8, 11, 14... (Common difference, d = 3)
• 10, 7, 4, 1, -2... (Common difference, d = -3)

The terms in an arithmetic sequence are often represented using the notation:

• a₁ (first term)
• a₂ (second term)
• a₃ (third term)
• ... and so on.

The general formula for the nth term of an arithmetic sequence is:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number
• d is the common difference

Methods for Finding the First Term (a₁)

Several methods exist for determining the first term of an arithmetic sequence, depending on the information provided. Let's explore each method in detail, providing illustrative examples for clarity.

Method 1: Using the General Formula and Known Terms

If you know the common difference (d) and any other term (a_n) in the sequence along with its position (n), you can directly use the general formula to find the first term (a₁). Simply rearrange the formula to solve for a₁:

a₁ = a_n - (n - 1)d

Example 1:

An arithmetic sequence has a common difference of 4, and its 5th term is 17. Find the first term.

• a_n = 17
• n = 5
• d = 4

Substituting these values into the rearranged formula:

a₁ = 17 - (5 - 1) * 4 = 17 - 16 = 1

Therefore, the first term is 1.

Example 2:

The 10th term of an arithmetic sequence is -22, and the common difference is -3. Find the first term.

• a_n = -22
• n = 10
• d = -3

a₁ = -22 - (10 - 1) * (-3) = -22 + 27 = 5

Therefore, the first term is 5.

Method 2: Using Two Known Terms and Their Positions

If you know two terms (a_m and a_n) and their positions (m and n) in the sequence, you can find the common difference (d) and then use it to find the first term (a₁).

The formula for the common difference using two known terms is:

d = (a_n - a_m) / (n - m)

Once you have the common difference, use the general formula (rearranged to solve for a₁) to find the first term.

Example 3:

The 3rd term of an arithmetic sequence is 11, and the 7th term is 23. Find the first term.

• a_m = 11 (m = 3)
• a_n = 23 (n = 7)

First, find the common difference:

d = (23 - 11) / (7 - 3) = 12 / 4 = 3

Now, use the rearranged general formula to find a₁, using either the 3rd or 7th term:

Using the 3rd term:

a₁ = 11 - (3 - 1) * 3 = 11 - 6 = 5

Using the 7th term:

a₁ = 23 - (7 - 1) * 3 = 23 - 18 = 5

Therefore, the first term is 5.

Method 3: Using the Sum of an Arithmetic Series and the Number of Terms

If you know the sum (S_n) of the first n terms of an arithmetic sequence and the number of terms (n), along with the common difference (d) or the last term (a_n), you can find the first term (a₁). The formula for the sum of an arithmetic series is:

S_n = (n/2) * [2a₁ + (n - 1)d] or S_n = (n/2) * (a₁ + a_n)

You'll need to rearrange these formulas to solve for a₁ depending on the information provided.

Example 4:

The sum of the first 6 terms of an arithmetic sequence is 99, and the common difference is 5. Find the first term.

• S_n = 99
• n = 6
• d = 5

Use the first sum formula:

99 = (6/2) * [2a₁ + (6 - 1) * 5]

99 = 3 * [2a₁ + 25]

33 = 2a₁ + 25

2a₁ = 8

a₁ = 4

Therefore, the first term is 4.

Example 5:

The sum of the first 8 terms of an arithmetic sequence is 120, and the 8th term is 21. Find the first term.

• S_n = 120
• n = 8
• a_n = 21

Use the second sum formula:

120 = (8/2) * (a₁ + 21)

120 = 4 * (a₁ + 21)

30 = a₁ + 21

a₁ = 9

Therefore, the first term is 9.

Addressing Complex Scenarios

Sometimes, the problem might not directly provide all the necessary information. You may need to combine multiple concepts and methods. For instance, you might need to solve a system of equations or use additional information to deduce the common difference or another term in the sequence before applying one of the methods above.

Example 6:

The sum of the first three terms of an arithmetic sequence is 18, and the third term is double the first term. Find the first term.

Here, we have two pieces of information:

1. S₃ = 18
2. a₃ = 2a₁

We can use the sum formula:

18 = (3/2) * (a₁ + a₃)

Substitute a₃ = 2a₁:

18 = (3/2) * (a₁ + 2a₁)

18 = (3/2) * 3a₁

18 = (9/2)a₁

a₁ = 18 * (2/9) = 4

Therefore, the first term is 4.

Practical Applications and Real-World Examples

Understanding arithmetic sequences and how to find the first term has applications in various fields:

• Finance: Calculating compound interest, loan repayments, or investment growth over time.
• Physics: Analyzing motion with constant acceleration, modeling projectile trajectories, or studying oscillations.
• Engineering: Designing structures, calculating material requirements, or modeling systems with linear growth or decay.
• Computer Science: Analyzing algorithms with linear complexity, managing data structures, or predicting program performance.

By mastering the techniques presented in this guide, you can tackle a wide range of problems involving arithmetic sequences and apply these mathematical principles to solve real-world challenges. Remember to always clearly identify the known variables and choose the most appropriate method based on the information given. Practice is key to becoming proficient in finding the first term and understanding the underlying principles of arithmetic sequences.