Solving for the First Term and Common Difference of an Arithmetic Progression

In the realm of sequences and series, an arithmetic progression (AP) is a fundamental concept. This article will guide you through solving a specific problem related to an arithmetic progression, where we need to find the first term and the common difference given the values of the 9th term and the 28th term.

Understanding the Problem

Given that the 9th term and the 28th term of an arithmetic progression are -5 and 128, respectively, our goal is to find the first term and the common difference of the sequence.

General Formula of an Arithmetic Progression

The nth term of an arithmetic progression (AP) can be given by the formula:

[a_n a_1 (n-1)d]

Where:

a_1 is the first term d is the common difference n is the term number

Solving for the Common Difference (d)

We are given two specific terms from the sequence:

Term 9: [a_9 -5]

Using the formula, we can express the 9th term as:

Equation 1: [a_1 8d -5]

Term 28: [a_{28} 128]

Similarly, the 28th term can be expressed as:

Equation 2: [a_1 27d 128]

To solve for d, we can subtract Equation 1 from Equation 2:

Equation 2 - Equation 1: [(a_1 27d) - (a_1 8d) 128 - (-5)]

This simplifies to:

19d 133

Solving for d: [d frac{133}{19} 7]

Solving for the First Term (a_1)

Now that we have the common difference d 7, we can substitute this value back into Equation 1 to find the first term a_1.

Equation 1: [a_1 8d -5]

Substituting d 7 into the equation:

Equation 1: [a_1 8(7) -5]

Equation 1: [a_1 56 -5]

Solving for a_1: [a_1 -5 - 56 -61]

Conclusion

The first term of the arithmetic progression is (-61) and the common difference is (7).

Values and Sequence

The sequence can be generated using the general term formula:

9th term (a_9): [-5]

28th term (a_{28}): [128]

The first few terms of the sequence are:

-61, -54, -47, -40, -33, -26, -19, -12, -5, 2, 9, 16, ...

The general term of the arithmetic sequence is given by:

[a_n -61 7(n-1) 7n - 68]

For different values of n, the terms of the sequence are:

When (n 1), (a_1 7(1) - 68 -61) When (n 2), (a_2 7(2) - 68 -54) When (n 3), (a_3 7(3) - 68 -47)

Key Takeaways

1. Common Difference (d): (7)

2. First Term (a_1): (-61)

3. General Term Formula: (a_n 7n - 68)