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:

  1. First Term (a): The starting number of the sequence.
  2. 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 = ...
  3. 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):

  1. Find the 10th term of the AP: 2, 7, 12, …
  2. Is the sequence 1, 4, 9, 16, … an AP? Justify your answer.
  3. If the first term of an AP is 10 and the common difference is -2, write the first four terms.
  4. In an AP, if the 3rd term is 18 and the 7th term is 30, find the common difference.
  5. Which term of the AP 21, 18, 15, … is -81?