Solving the First Term of an Arithmetic Progression

In this article, we will delve into a complex problem involving an arithmetic progression (AP). We will walk through the detailed steps to find the first term of an AP, given the 10th term is (frac{13}{5}) times the 4th term and the common difference is 4. This problem requires a solid understanding of arithmetic sequences and algebraic manipulation.

Problem Statement

We need to determine the value of the first term (a) for an arithmetic progression (AP) where:

The common difference (d 4). The 10th term is (frac{13}{5}) times the 4th term.

Let's define the general term of an AP as:

[t_n a (n-1)d]

Solving for the First Term (a)

We know the 10th term and the 4th term in terms of (a) and (d) are:

[t_{10} a 9d a 36]

[t_4 a 3d a 12]

According to the problem statement, the 10th term is (frac{13}{5}) times the 4th term:

[a 36 frac{13}{5}(a 12)]

Let's solve this equation to find the value of (a).

First, simplify the equation: [a 36 frac{13a}{5} frac{156}{5}] Multiply every term by 5 to clear the fraction: [5a 180 13a 156] Group the terms to isolate (a): [180 - 156 13a - 5a] [24 8a] Solve for (a): [a frac{24}{8} 3]

Conclusion

The first term of the AP is 3.

The Arithmetic Sequence

The terms of the arithmetic sequence can be calculated as:

First term: (a_1 3) Second term: (a_2 7) Third term: (a_3 11) Fourth term: (a_4 15) ...

The general term of this sequence is:

[a_n 4n - 1]

Intermediate Steps for Clarification

To further clarify the steps, let's solve the equation in two intermediary steps:

Calculate the 4th term: [t_4 a 12] Given (13/5 t_4), substitute: [t_{10} 13/5 (a 12)] Which simplifies to: [a 36 13/5 (a 12)] Then, solve as above: [a 3]

Therefore, the first term of the AP is (a 3).

Example: The arithmetic sequence is: (3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, ...)