Introduction to Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a special type of sequence where the difference between consecutive terms is constant, known as the common difference, denoted by d. This constant difference, when added to any term, results in the next term of the sequence.

Solving for the Common Difference and First Term

Given the 10th term and the 18th term of an AP, we will determine the first three terms of the sequence. Let's follow the steps to find the values.

Expressing the Term Formula

The formula for the nth term of an AP is:

an a (n - 1) d

Where a is the first term, d is the common difference, and n is the term number.

Setting Up Equations

We know that the 10th term and 18th term of the AP are given as:

The 10th term 5 The 18th term 72

Using the term formula, we can write two equations:

10th term: a 9d 5 18th term: a 17d 72

Solving the Equations

To find the common difference d, we subtract the first equation from the second:

a 17d - (a 9d) 72 - 5

8d 67

d 67/8

Now, substituting the value of d back into the first equation to find the first term a:

a 9(67/8) 5

a 603/8 5

a 5 - 603/8

a -563/8

Finding the First Three Terms

Knowing a and d , we can now find the first three terms:

First term: a -563/8 Second term: a d -563/8 67/8 -496/8 -62 Third term: a 2d -563/8 2(67/8) -563/8 134/8 -429/8

Therefore, the first three terms of the given AP are -563/8, -62, and -429/8 respectively.

Conclusion

Understanding the properties of an arithmetic progression and applying them through equations is essential for solving problems related to sequences and series. The process involves substituting the known terms into the formula, solving the equations, and then finding the required terms.