Math's: Arithmetic Progressions: Meaning, Term and Sum of First N Terms

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Arithmetic Progressions- Meaning

Arithmetic Progression (AP) is a type of series in which each term except the first is obtained by adding a fixed number (positive or negative) to the previous term. The first term is denoted by and the common difference is denoted by

Thus, the standard form of an Arithmetic Progression would be:

Examples-

(i)

Arithmetic Progressions

Arithmetic Progressions

(ii)

(iii)

1. Which of the following series are Arithmetic Progressions? In

case of find their respective first terms and common differences.

(i)

(ii)

(iii)

Sol:

(i) We see that in

. The difference between the two successive terms are not equal. So, the series is not AP.

(ii)

. The difference between the two consecutive terms are equal. So, the series is AP where the first term is and the common difference is .

(iii)

The difference between the two consecutive terms are equal. So, the series is AP where the first term is and the common difference is .

General (Nth) Term of an AP

General (nth) Term of an AP is denoted by is worked out by the following formula-

Example- Find the tenth term of the AP-

Sol- Here,

Tenth Term

Sum of First in Terms of an AP

The sum of first in terms of an AP is calculated from the following formula-

Where denotes sum of n terms is the first term and denotes the common difference.

Sometimes, nth term is named as last term and is denoted by . In such a case-

Example- Find the sum of first 20 even numbers.

Sol- The series for first 20 even numbers would be-

Here,

The sum of first 20 even numbers is

2. In an A.P., the sum of first three numbers is 15 and their product is 45. Find the three numbers.

Solution- Let the three numbers be and respectively.

As per the question,

By substituting the value of from eq (1) to eq (2) we get,

Putting the value of in the equation (1), we get-

Thus, the numbers are and-

and

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