Solving Arithmetic Progression Problems: Finding the 14th Term

An arithmetic progression (AP) is a sequence of numbers such that the difference between any two successive members is constant. This constant difference is known as the common difference (CD). This article will guide you through the process of finding the 14th term of an arithmetic progression given the 2nd and 8th terms.

Given Information

In this problem, we are given the 2nd term (T2) as 17 and the 8th term (T8) as -1. We need to find the 14th term (T14).

Formulas and Steps

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

T_n a (n-1)d

Where:

a is the first term of the progression, d is the common difference.

Step 1: Express the Given Terms Using the n-th Term Formula

For the 2nd term:

T_2 a (2-1)d a d 17

For the 8th term:

T_8 a (8-1)d a 7d -1

Step 2: Solve for the Common Difference (d)

We now have two equations:

a d 17 ... 1

a 7d -1 ... 2

Subtract equation 1 from equation 2:

(a 7d) - (a d) -1 - 17

Simplifying, we get:

6d -18

Therefore:

d -18 / 6 -3

Step 3: Solve for the First Term (a)

Substitute the value of d into equation 1:

a - 3 17

Therefore:

a 20

Step 4: Find the 14th Term (T14)

Using the general formula for the n-th term:

T_14 a (14-1)d 20 13(-3) 20 - 39 -19

Thus, the 14th term of the arithmetic progression is -19.

Conclusion

We have successfully found the 14th term of the given arithmetic progression. The key is to use the general formula for the n-th term and manipulate the given terms to solve for the common difference and the first term. Once we have these values, we can easily find any term in the sequence.

Keywords: arithmetic progression, common difference, 14th term

Related Article Tags: arithmetic progression, sequence, common difference, term calculation