Class 10 Mathematics Notes Chapter 5 (Chapter 5) – Examplar Problems (English) Book

Alright class, let's focus on Chapter 5, Arithmetic Progressions (AP), from your NCERT Exemplar. This is a crucial topic, not just for your board exams but also frequently tested in various government competitive examinations. Pay close attention to the concepts and problem-solving techniques.

Chapter 5: Arithmetic Progressions (AP) - Detailed Notes for Competitive Exams

1. What is an Arithmetic Progression?
An Arithmetic Progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first term.

• Sequence: A set of numbers arranged in a definite order.
• Progression: A sequence following a specific pattern.
• Example: 2, 5, 8, 11, ... (Each term is obtained by adding 3 to the previous one)
• Example: 100, 90, 80, 70, ... (Each term is obtained by adding -10 to the previous one)

2. Key Terms & Notation:

• First Term (a or a₁): The starting term of the AP.
• Common Difference (d): The fixed number added to get successive terms. It can be positive, negative, or zero.
  • d = a₂ - a₁ = a₃ - a₂ = ... = aₙ - aₙ₋₁
• nth Term (aₙ or Tₙ): The term at the nth position in the sequence. Also called the General Term.
• Sum of First n Terms (Sₙ): The sum of the terms from a₁ to aₙ.

3. Formulas (Memorize These!)

• (a) nth Term (General Term):
  aₙ = a + (n - 1)d

  • aₙ: The value of the term you want to find.
  • a: The first term.
  • n: The position of the term in the sequence.
  • d: The common difference.
  • Use Case: Finding the 20th term, finding which term equals a certain value, etc.
  • If the AP is finite, the last term 'l' is often used instead of aₙ, so l = a + (n - 1)d.

• (b) Sum of First n Terms:

  • Formula 1 (When 'a', 'n', and 'd' are known):
    Sₙ = n/2 [2a + (n - 1)d]
  • Formula 2 (When 'a', 'n', and the last term 'aₙ' or 'l' are known):
    Sₙ = n/2 [a + aₙ] or Sₙ = n/2 [a + l]
  • Use Case: Finding the sum of the first 15 terms, finding how many terms add up to a certain sum, etc.

4. Important Properties of APs:

• Checking for AP: A sequence a₁, a₂, a₃, ... is an AP if and only if the difference between consecutive terms (aₖ₊₁ - aₖ) is constant for all k ≥ 1.
• Effect of Adding/Subtracting: If a constant 'k' is added to or subtracted from each term of an AP, the resulting sequence is also an AP with the same common difference.
• Effect of Multiplying/Dividing: If each term of an AP is multiplied or divided by a non-zero constant 'k', the resulting sequence is also an AP with the common difference multiplied or divided by 'k' (i.e., kd or d/k).
• Three terms in AP: If a, b, c are in AP, then 2b = a + c. 'b' is called the Arithmetic Mean (AM) of 'a' and 'c'. b = (a + c) / 2.
• Relationship between Sₙ and aₙ: The nth term of an AP is the difference between the sum of the first n terms and the sum of the first (n-1) terms.
  aₙ = Sₙ - Sₙ₋₁ (This formula is valid for n > 1)
  • Note: The first term a₁ = S₁. This is very useful when the formula for Sₙ is given, and you need to find the AP or a specific term.

5. Selecting Terms in an AP (Useful for Problem Solving):

When the sum of a certain number of terms in an AP is given, choosing the terms strategically simplifies calculations:

• 3 terms: a - d, a, a + d (Sum = 3a)
• 4 terms: a - 3d, a - d, a + d, a + 3d (Sum = 4a; Common difference here is 2d)
• 5 terms: a - 2d, a - d, a, a + d, a + 2d (Sum = 5a)

6. Common Problem Types in Competitive Exams (Based on Exemplar):

• Finding 'd' when terms are given in algebraic form (e.g., find 'k' if k, 2k+1, 3k+2 are in AP).
• Finding a specific term (e.g., 15th term) or the number of terms ('n').
• Finding the sum of 'n' terms or finding 'n' when the sum is given.
• Problems where Sₙ is given as a quadratic expression in 'n'. Use aₙ = Sₙ - Sₙ₋₁ to find the terms or 'd'. Remember that 'd' will be twice the coefficient of n².
• Problems involving relationships between terms (e.g., if the mth term is 1/n and the nth term is 1/m, find the (mn)th term).
• Word problems involving installments, savings patterns, logs stacked in rows, etc., where the underlying pattern is an AP.
• Finding the sum of integers divisible by a certain number within a given range (e.g., sum of integers between 100 and 500 divisible by 7).
• Determining if a given number is a term in a specific AP. (Find 'n' using aₙ = a + (n-1)d. If 'n' is a positive integer, the number is a term).

7. Key Tips for Exams:

• Read the question carefully to identify 'a', 'd', 'n', 'aₙ', or 'Sₙ'.
• Double-check your calculations, especially with negative signs in 'd'.
• Remember the specific formulas and when to use each version of the Sₙ formula.
• Practice the aₙ = Sₙ - Sₙ₋₁ relationship.
• For MCQ questions, sometimes substituting the options back into the conditions can be faster.
• Understand the difference between the 'nth term' and the 'sum of n terms'.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, similar to what you might encounter:

1. Which term of the AP: 21, 18, 15, ... is -81?
   (A) 33rd
   (B) 34th
   (C) 35th
   (D) 36th

2. If the 2nd term of an AP is 13 and the 5th term is 25, what is its 7th term?
   (A) 30
   (B) 33
   (C) 37
   (D) 38

3. The sum of the first 16 terms of the AP: 10, 6, 2, ... is:
   (A) -320
   (B) 320
   (C) -352
   (D) -400

4. If the first term of an AP is -5 and the common difference is 2, then the sum of the first 6 terms is:
   (A) 0
   (B) 5
   (C) 6
   (D) 15

5. The sum of the first five multiples of 3 is:
   (A) 45
   (B) 55
   (C) 65
   (D) 75

6. If the numbers n – 2, 4n – 1 and 5n + 2 are in AP, then the value of n is:
   (A) 1
   (B) 2
   (C) -1
   (D) -2

7. The 10th term from the end of the AP: 4, 9, 14, ..., 254 is:
   (A) 209
   (B) 205
   (C) 214
   (D) 213

8. If the sum of the first 'n' terms of an AP is given by Sₙ = 3n² + 5n, then its common difference is:
   (A) 5
   (B) 6
   (C) 7
   (D) 8

9. The sum of first n positive integers is given by:
   (A) n(n+1)/2
   (B) n(n-1)/2
   (C) n²(n+1)/2
   (D) n(n+1)

10. In an AP, if d = -4, n = 7, aₙ = 4, then 'a' is equal to:
    (A) 6
    (B) 7
    (C) 20
    (D) 28

Answer Key for MCQs:

1. (C) 35th
2. (B) 33
3. (A) -320
4. (A) 0
5. (A) 45
6. (A) 1
7. (A) 209 (Hint: Consider the AP reversed or use aₙ from the end = l - (n-1)d)
8. (B) 6 (Hint: Find a₁=S₁, a₂=S₂-S₁, then d=a₂-a₁)
9. (A) n(n+1)/2
10. (D) 28

Study these notes thoroughly, practice problems from the Exemplar book, and focus on understanding the application of these formulas. Good luck with your preparation!