Class 10 Mathematics Notes Chapter 5 (Arithmetic progressions) – Mathematics Book

Class!, Read notes of Chapter 5: Arithmetic Progressions (APs). This is a fundamental topic, and understanding it well is crucial for many government exams where quantitative aptitude is tested. Pay close attention to the definitions, formulas, and how they are applied.

Chapter 5: Arithmetic Progressions (APs) - Detailed Notes

1. Introduction & Definition:

• An Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant.

• This constant difference is called the common difference (d).

• Each number in the sequence is called a term.

• Examples:

  • 2, 4, 6, 8, ... (Here, the first term is 2, and the common difference is 4 - 2 = 2)
  • 100, 90, 80, 70, ... (Here, the first term is 100, and the common difference is 90 - 100 = -10)
  • 5, 5, 5, 5, ... (Here, the first term is 5, and the common difference is 5 - 5 = 0)

• The common difference 'd' can be positive, negative, or zero.

2. General Form of an AP:

• If the first term is denoted by 'a' and the common difference by 'd', then the AP can be written as:
  a, a + d, a + 2d, a + 3d, ...
• The first term is a_1 = a
• The second term is a_2 = a + d
• The third term is a_3 = a + 2d
• And so on...

3. Finding the Common Difference (d):

• To find the common difference, subtract any term from its succeeding term.
• d = a_2 - a_1 = a_3 - a_2 = ... = a_k+1 - a_k
• If the difference between consecutive terms is not constant, the sequence is not an AP.

4. The nth Term (General Term) of an AP:

• Sometimes we need to find a specific term in the AP, like the 10th term, 50th term, etc., without listing all the terms.

• The formula for the nth term (a_n) of an AP is:
  a_n = a + (n - 1)d
  Where:

  • a_n = the nth term
  • a = the first term
  • n = the position of the term in the sequence
  • d = the common difference

• This formula is extremely important. It helps find any term if 'a', 'd', and 'n' are known. It can also be used to find 'n', 'a', or 'd' if other values are given.

• a_n is also sometimes denoted by l if it is the last term of a finite AP.

• Example: Find the 15th term of the AP: 3, 7, 11, 15, ...

  • Here, a = 3, d = 7 - 3 = 4, n = 15
  • a_15 = a + (15 - 1)d = 3 + (14) * 4 = 3 + 56 = 59
  • So, the 15th term is 59.

5. Sum of the First n Terms of an AP:

• Often, we need to find the sum of a certain number of terms in an AP.

• The formula for the sum of the first n terms (S_n) is:
  S_n = n/2 [2a + (n - 1)d]
  Where:

  • S_n = the sum of the first n terms
  • n = the number of terms
  • a = the first term
  • d = the common difference

• Alternative Formula for S_n: If the first term (a) and the last term (a_n or l) are known, we can use a simpler formula:
  S_n = n/2 [a + l] or S_n = n/2 [a + a_n]
  (This is derived from the first formula since a_n = a + (n-1)d, so 2a + (n-1)d = a + [a + (n-1)d] = a + a_n)

• When to use which formula?

  • Use S_n = n/2 [2a + (n - 1)d] when you know a, d, and n.
  • Use S_n = n/2 [a + l] when you know a, l (the last term), and n.

• Example: Find the sum of the first 20 terms of the AP: 1, 4, 7, 10, ...

  • Here, a = 1, d = 4 - 1 = 3, n = 20
  • Using S_n = n/2 [2a + (n - 1)d]
  • S_20 = 20/2 [2(1) + (20 - 1) * 3]
  • S_20 = 10 [2 + (19) * 3]
  • S_20 = 10 [2 + 57]
  • S_20 = 10 [59] = 590
  • The sum of the first 20 terms is 590.

6. Important Properties & Tips:

• If a, b, c are in AP, then 2b = a + c (The middle term is the arithmetic mean of the other two). This means b - a = c - b.
• To check if a sequence is an AP, calculate the difference between consecutive terms. If it's constant, it's an AP.
• Problems often involve finding a_n, S_n, n, a, or d when some of these are given. Use the two main formulas (a_n and S_n) and solve the resulting equations.
• Word problems involving APs often deal with quantities increasing or decreasing by a fixed amount per period (e.g., salary increments, savings patterns, production levels). Identify a, d, and n from the problem statement.

Practice MCQs for Government Exams

Here are 10 Multiple Choice Questions based on Arithmetic Progressions:

1. Which of the following sequences is an Arithmetic Progression?
   (A) 1, 3, 6, 10, ...
   (B) 3, 6, 12, 24, ...
   (C) -10, -6, -2, 2, ...
   (D) 1, 11, 21, 30, ...
   Answer: (C) (Common difference is -6 - (-10) = 4)

2. What is the common difference of the AP: 1/2, 1, 3/2, 2, ...?
   (A) 1/2
   (B) 1
   (C) -1/2
   (D) 0
   Answer: (A) (d = 1 - 1/2 = 1/2)

3. The first term of an AP is 5 and the common difference is -3. What is the 10th term?
   (A) -22
   (B) -25
   (C) 32
   (D) 22
   Answer: (A) (a_10 = a + (10-1)d = 5 + 9*(-3) = 5 - 27 = -22)

4. How many terms are there in the AP: 7, 13, 19, ..., 205?
   (A) 33
   (B) 34
   (C) 35
   (D) 36
   Answer: (B) (a=7, d=6, a_n=205. Use a_n = a + (n-1)d => 205 = 7 + (n-1)6 => 198 = (n-1)6 => 33 = n-1 => n=34)

5. The 4th term of an AP is 11 and the 10th term is 29. What is the common difference?
   (A) 2
   (B) 3
   (C) 4
   (D) 5
   Answer: (B) (a_10 - a_4 = (10-4)d => 29 - 11 = 6d => 18 = 6d => d=3)

6. What is the sum of the first 10 terms of the AP: 2, 7, 12, ...?
   (A) 245
   (B) 255
   (C) 265
   (D) 235
   Answer: (A) (a=2, d=5, n=10. S_10 = 10/2 [2(2) + (10-1)5] = 5 [4 + 9*5] = 5 [4 + 45] = 5 * 49 = 245)

7. The first term of an AP is 1 and the last term is 11. If the sum of its terms is 36, what is the number of terms?
   (A) 5
   (B) 6
   (C) 7
   (D) 8
   Answer: (B) (a=1, l=11, S_n=36. Use S_n = n/2 [a + l] => 36 = n/2 [1 + 11] => 36 = n/2 * 12 => 36 = 6n => n=6)

8. If the sum of the first n terms of an AP is given by S_n = 3n^2 + 5n, what is the common difference?
   (A) 4
   (B) 6
   (C) 8
   (D) 10
   Answer: (B) (a_1 = S_1 = 3(1)^2 + 5(1) = 8. S_2 = 3(2)^2 + 5(2) = 12 + 10 = 22. a_2 = S_2 - S_1 = 22 - 8 = 14. d = a_2 - a_1 = 14 - 8 = 6)

9. What is the sum of the first 10 positive multiples of 6?
   (A) 300
   (B) 330
   (C) 360
   (D) 390
   Answer: (B) (The AP is 6, 12, 18, ... . We need the sum of the first 10 terms. a=6, d=6, n=10. S_10 = 10/2 [2(6) + (10-1)6] = 5 [12 + 9*6] = 5 [12 + 54] = 5 * 66 = 330)

Revise these concepts thoroughly. Practice identifying a, d, n, a_n, and S_n in different problems and apply the formulas correctly. Good luck with your preparation!