Calculating the First Term and Common Difference of an Arithmetic Progression

Arithmetic progressions are a fundamental concept in mathematics, often appearing in various contexts, from basic algebra to complex problem-solving scenarios. In this article, we will explore how to find the first term and the common difference of an arithmetic progression given specific terms. Let's take a deep dive into the calculations involved.

Understanding the Problem

Given two terms of an arithmetic progression (AP), the 6th term is (2 frac{1}{2}) and the 10th term is (2 frac{5}{6}). Our goal is to determine the first term and common difference of the arithmetic progression.

Solving the Problem

Step 1: Define the Terms

The general formula for the (n)th term of an arithmetic progression is given by:

$$a_n a_m (n - m)d$$

Let's denote the first term by (a_1) and the common difference by (d).

Step 2: Set Up the Equations

Using the given information:

$$a_{10} a_6 4d 2 frac{5}{6}$$

However, we know that:

$$a_6 a_1 5d 2 frac{1}{2}$$

Substitute (a_6) into the equation for (a_{10}):

$$a_{10} a_6 4d (a_1 5d) 4d a_1 9d 2 frac{5}{6}$$

Now, we have the equation:

$$a_1 9d 2 frac{5}{6}$$

Step 3: Convert Mixed Fractions to Improper Fractions

Convert (2 frac{5}{6}) to an improper fraction:

$$2 frac{5}{6} frac{17}{6}$$

And (2 frac{1}{2}) to an improper fraction:

$$2 frac{1}{2} frac{5}{2}$$

Step 4: Solve for the Common Difference (d)

Using the equation:

$$a_1 9d frac{17}{6}$$

And knowing:

$$a_1 5d frac{5}{2}$$

Subtract these two equations to isolate (d):

$$a_1 9d - (a_1 5d) frac{17}{6} - frac{5}{2}$$ $$4d frac{17}{6} - frac{15}{6} frac{2}{6} frac{1}{3}$$

Thus,

$$d frac{1}{12}$$

Step 5: Solve for the First Term (a_1)

Substitute (d) into the equation:

$$a_1 9d frac{17}{6}$$

Substitute (d frac{1}{12}):

$$a_1 9 times frac{1}{12} frac{17}{6}$$ $$a_1 frac{9}{12} frac{17}{6}$$ $$a_1 frac{3}{4} frac{17}{6}$$

Convert (frac{3}{4}) to a fraction with a denominator of 12:

$$a_1 frac{9}{12} frac{34}{12}$$

Subtract (frac{9}{12}) from both sides:

$$a_1 frac{34}{12} - frac{9}{12} frac{25}{12}$$

Conclusion

The first term (a_1) of the arithmetic progression is (frac{25}{12}) or (2 frac{1}{12}), and the common difference (d) is (frac{1}{12}).

References

For further reading and practice on arithmetic progression, you may refer to:

MathWorld: Khan Academy:

Understanding arithmetic progressions is crucial for solving various mathematical problems, particularly in algebra and number theory.