NCERT| RBSE, CBSE | CLASS TENTH, SUBJECT MATHS, UNIT SECOND, ARITHMETIC PROGRESSIONS

Arithmetic Progressions: 8 Questions and Answers

An arithmetic progression (AP) is a sequence of numbers where each term is equal to the previous term plus a constant difference. This constant difference is called the common difference (d).

1. What is the nth term of an arithmetic progression?

The nth term (a_n) of an AP can be found using the formula:

a_n = a_1 + (n – 1)d

where:

• a_1 is the first term
• n is the term number
• d is the common difference

2. What is the sum of the first n terms of an AP?

The sum of the first n terms of an AP can be found using the formula:

S_n = n/2 (a_1 + a_n)

or alternatively:

S_n = n/2 (2a_1 + (n – 1)d)

where:

• n is the number of terms
• a_1 is the first term
• a_n is the nth term
• d is the common difference

3. How can we identify an AP?

An AP can be identified by the following characteristics:

• The difference between any two consecutive terms is constant.
• The sequence can be represented by a formula like a_n = a_1 + (n – 1)d.
• Plotting the terms on a graph results in a straight line.

4. What are some real-world examples of APs?

• The number of steps you take while climbing stairs.
• The cost of renting a car for increasing days.
• The temperature increasing at a constant rate.
• The seating arrangement in a movie theater.

5. Can you give me an example of finding the nth term of an AP?

In an AP, the first term is 3 and the common difference is 5. What is the 10th term?

Using the formula:

a_10 = 3 + (10 – 1) * 5 = 53

Therefore, the 10th term in the AP is 53.

6. Can you give me an example of finding the sum of the first n terms of an AP?

In an AP, the first term is 2 and the common difference is 4. What is the sum of the first 5 terms?

Using the formula:

S_5 = 5/2 (2 + (5 – 1) * 4) = 5/2 * 22 = 55

Therefore, the sum of the first 5 terms in the AP is 55.

7. What happens if the common difference is negative?

If the common difference is negative, the sequence decreases instead of increasing. The formulas for the nth term and the sum of the first n terms still apply, but the value of d will be negative.

8. What are some other ways to analyze APs?

• Finding the explicit formula of the AP: Expressing the nth term as a function of n.
• Finding missing terms: Filling in the gaps in the sequence based on the known terms and common difference.
• Solving problems involving APs: Applying the formulas and properties of APs to real-world situations.

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