Chapter 5: Arithmetic Progressions (समान्तर श्रेढ़ी)

Chapter 5: Arithmetic Progressions (समान्तर श्रेढ़ी)

Class 10 – CBSE Basic Maths

Quick Summary for Exam: For Basic Maths, focus on memorizing two main formulas: the “nth term” (an) and the “Sum of n terms” (Sn). Most questions are direct formula-based.

1. What is an AP? (AP क्या है?)

Definition:

• English: An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term (except the first term).
• Hindi: ऐसी श्रेणी जिसमें हर अगला पद, पिछले पद में एक निश्चित संख्या जोड़ने पर प्राप्त होता है।

Key Terms:

1. First Term (प्रथम पद): Denoted by ‘a’.
2. Common Difference (सार्व अंतर): Denoted by ‘d’.
   • It can be positive, negative, or zero.
   • Formula: d = Second term – First term (a₂ – a₁)

Example: AP: 2, 5, 8, 11…

• First term (a) = 2
• Common Difference (d) = 5 – 2 = 3

2. General Form of an AP (AP का व्यापक रूप)

If ‘a’ is the first term and ‘d’ is the common difference, the AP is written as:

a, a+d, a+2d, a+3d…

• 1st term = a
• 2nd term = a + d
• 3rd term = a + 2d
• 4th term = a + 3d

3. Finding the nth Term (AP का nवाँ पद)

This formula is used to find any specific term (like the 10th term or 50th term) or to find the total number of terms.

Formula: an = a + (n – 1)d

• an = nth term (nवाँ पद)
• a = First term (प्रथम पद)
• n = Number of terms (पदों की संख्या)
• d = Common difference (सार्व अंतर)

Example Question 1:

Find the 10th term of the AP: 2, 7, 12… (AP: 2, 7, 12… का 10वाँ पद ज्ञात कीजिए)

Solution:

• a = 2
• d = 7 – 2 = 5
• n = 10

Formula: a₁₀ = a + (10-1)d a₁₀ = 2 + 9 × 5 a₁₀ = 2 + 45 a₁₀ = 47

4. Sum of First n Terms (प्रथम n पदों का योग)

This formula is used when you need to add up the terms of an AP (e.g., “Find the sum of the first 20 terms”).

Formula 1 (General): Sn = n/2 × [2a + (n – 1)d]

Formula 2 (If Last Term ‘l’ is known): Sn = n/2 × (a + l)

Example Question 2:

Find the sum of the first 22 terms of the AP: 8, 3, -2… (AP: 8, 3, -2… के प्रथम 22 पदों का योग ज्ञात कीजिए)

Solution:

• a = 8
• d = 3 – 8 = -5 (Note: d is negative here)
• n = 22

Formula: S₂₂ = (22/2) × [2(8) + (22-1)(-5)] S₂₂ = 11 × [16 + 21(-5)] S₂₂ = 11 × [16 – 105] S₂₂ = 11 × [-89] S₂₂ = -979

5. Important Tips for Basic Maths (महत्वपूर्ण सुझाव)

1. Finding ‘n’: Often the question asks “Which term of the AP is 78?”.
   • Here, assume an = 78.
   • Use the formula 78 = a + (n-1)d and solve for n.
2. Check ‘d’: Always calculate ‘d’ carefully.
   • Wrong: d = 8 – 3 = 5
   • Right: d = 3 – 8 = -5 (Always Second Term – First Term)
3. Sum of Positive Integers:
   • Sum of first n positive numbers = n(n+1)/2

6. Cheatsheet (सूत्रों का सारांश)

English	Hindi	Formula
nth Term	nवाँ पद	an = a + (n-1)d
Sum of n terms	n पदों का योग	Sn = n/2 × [2a + (n-1)d]
Sum (Last term known)	योग (अंतिम पद ज्ञात हो)	Sn = n/2 × (a + l)
Common Difference	सार्व अंतर	d = a₂ – a₁