Introduction

Arithmetic progressions (APs) are commonly used in mathematical problems to find specific terms within a sequence. In this article, we will explore how to find the 13th term of an AP given the relationship between the 5th and 8th terms. This problem is not only theoretical; it also has applications in various real-world scenarios, such as finance, engineering, and data analysis.

Understanding the Problem

We are given that five times the 5th term of an arithmetic progression is equal to eight times its 8th term. Our task is to find the value of the 13th term.

Setting Up the Equations

Let the first term of the arithmetic progression be (a) and the common difference be (d).

The General Term of an AP

The n-th term of an AP can be expressed as:

(T_n a(n-1)d)

Solving the Problem

Given that five times the 5th term is equal to eight times the 8th term, we can write:

(5 cdot T_5 8 cdot T_8)

Calculating (T_5) and (T_8):

(T_5 a 4d)

(T_8 a 7d)

Formulating the Equation

Substituting these into the given equation, we get:

(5(a 4d) 8(a 7d))

Expanding both sides:

(5a 20d 8a 56d)

Solving for (a) and (d)

Rearranging the equation:

(5a - 8a 20d - 56d 0)

This simplifies to:

(-3a - 36d 0)

Further simplifying:

(-3a 36d)

(-a 12d)

(a -12d)

Finding the 13th Term

Now we can find the 13th term (left(T_{13}right)):

(T_{13} a 12d)

Substituting (a -12d):

(T_{13} -12d 12d 0)

Hence, the 13th term of the AP is:

0

Conclusion

The 13th term of the AP is 0. This solution demonstrates the power of algebraic manipulation in solving complex AP problems. Understanding such mathematical concepts is crucial in various fields, including computer science, finance, and engineering, where arithmetic sequences play a significant role.