Arithmetic Progressions (AP): Motivation MCQs Quiz | Class 10
This quiz covers Class X Mathematics (Code 041), Unit II: Algebra, focusing on the topic Arithmetic Progressions (AP): Motivation. It includes questions on pattern recognition and sequences as AP. Submit your answers at the end and download a PDF of your results.
Understanding Arithmetic Progressions (AP)
An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by ‘d’. APs are widely observed in various real-life patterns and form a fundamental concept in mathematics.
Key Concepts of an AP:
- First Term (a): The starting number of the sequence.
- Common Difference (d): The constant value added to each term to get the next term. It can be positive (for increasing APs), negative (for decreasing APs), or zero (for constant APs).
d = a2 - a1 = a3 - a2 = ... - General Term (an): The formula to find any nth term of an AP is given by:
an = a + (n - 1)d
where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the position of the term.
Pattern Recognition leading to APs:
Many natural and man-made patterns follow an AP. For example:
- The number of seats in rows of an auditorium, where each subsequent row has a fixed number of seats more than the previous one.
- The total amount of loan repaid if a fixed amount is paid each month.
- The height of a plant growing by a fixed number of centimeters each week.
Example Table: Identifying an AP
| Term Number (n) | Term (an) | Difference from previous term (an – a(n-1)) |
|---|---|---|
| 1 | 4 | – |
| 2 | 9 | 9 – 4 = 5 |
| 3 | 14 | 14 – 9 = 5 |
| 4 | 19 | 19 – 14 = 5 |
Since the difference between consecutive terms is consistently 5, this is an AP with a first term a = 4 and a common difference d = 5.
Quick Revision Points:
- An AP has a constant difference between consecutive terms.
- This constant difference is called the common difference (d).
- d can be positive, negative, or zero.
- The general form of an AP is
a, a+d, a+2d, a+3d, ... - The nth term of an AP is
an = a + (n - 1)d.
Practice Questions (Not part of the quiz):
- Find the 10th term of the AP: 2, 7, 12, …
- Is the sequence 1, 4, 9, 16, … an AP? Justify your answer.
- If the first term of an AP is 10 and the common difference is -2, write the first four terms.
- In an AP, if the 3rd term is 18 and the 7th term is 30, find the common difference.
- Which term of the AP 21, 18, 15, … is -81?
