📘 Arithmetic and Geometric Progression (AG & GP) / अंकगणित और ज्यामितीय प्रगति

Goal of this chapter: By the end of this lesson, you will feel confident with the key NDA-level concepts of 📘 Arithmetic and Geometric Progression (AG & GP) / अंकगणित और ज्यामितीय प्रगति and be able to solve exam-style questions on your own.

हिंदी Note: इस chapter को आराम से पढ़िए – examples और छोटे-छोटे अभ्यास के साथ। एक बार समझ आ गया तो NDA में 📘 Arithmetic and Geometric Progression (AG & GP) / अंकगणित और ज्यामितीय प्रगति के सवाल आपके लिए बहुत आसान हो जाएंगे 😄

1. Arithmetic Progression (AP) / अंकगणित प्रगति (AP)

Definition: A sequence of numbers where the difference between consecutive terms is constant.
परिभाषा: संख्याओं की ऐसी श्रंखला जिसमें प्रत्येक क्रमागत पद के बीच का अंतर समान होता है।

nth term (nवाँ पद): a_n = a + (n - 1)d
nवाँ पद: a_n = a + (n - 1)d

Sum of first n terms / पहले n पदों का योग:
S_n = n/2 × [2a + (n - 1)d] or S_n = n/2 × (a + l)
S_n = n/2 × [2a + (n - 1)d] या S_n = n/2 × (a + l)

2. Geometric Progression (GP) / ज्यामितीय प्रगति (GP)

Definition: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
परिभाषा: एक ऐसी श्रंखला जिसमें प्रत्येक पद उसके पूर्ववर्ती पद का एक निश्चित अनुपात होता है।

nth term (nवाँ पद): a_n = a × r^n-1
nवाँ पद: a_n = a × r^n-1

Sum of first n terms / पहले n पदों का योग:
S_n = a(1 - rⁿ) / (1 - r), if r ≠ 1
S_n = a(1 - rⁿ) / (1 - r), यदि r ≠ 1 हो

Sum to infinity (when |r| < 1) / अनंत पदों का योग (जब |r| < 1):
S = a / (1 - r)
S = a / (1 - r)

Hint-Based Questions (With Step-by-Step Hints) / संकेतों के साथ प्रश्न

Q: Find the 10th term of the AP: 3, 7, 11, ...

Hint / संकेत: Use the formula: aₙ = a + (n-1)d. Here, a = 3, d = 4, n = 10

Answer / उत्तर: a₁₀ = 3 + (10-1)×4 = 3 + 36 = 39

Q: What is the sum of the first 20 terms of the AP: 2, 5, 8, ...?

Hint / संकेत: Use Sₙ = n/2 [2a + (n-1)d]. a = 2, d = 3, n = 20

Answer / उत्तर: S₂₀ = 20/2 [2×2 + 19×3] = 10 × [4 + 57] = 10 × 61 = 610

Q: Find the 5th term of a GP with a = 2 and r = 3.

Hint / संकेत: Use aₙ = a·rⁿ⁻¹. a = 2, r = 3, n = 5

Answer / उत्तर: a₅ = 2·3⁴ = 2·81 = 162

Q: Sum of first 4 terms of GP: 1, 2, 4, 8.

Hint / संकेत: Use Sₙ = a(rⁿ - 1)/(r - 1). a = 1, r = 2, n = 4

Answer / उत्तर: S₄ = (2⁴ - 1)/(2 - 1) = (16 - 1)/1 = 15

Q: Which term of AP: 5, 10, 15, ... is 100?

Hint / संकेत: Use aₙ = a + (n-1)d. 100 = 5 + (n-1)×5 ⇒ (n-1) = 19 ⇒ n = 20

Answer / उत्तर: 20th term is 100

Practice Questions / अभ्यास प्रश्न

Q: Find the 12th term of the AP: 1, 4, 7, ...

Answer / उत्तर: a₁₂ = 34

Q: What is the sum of first 15 terms of AP: 6, 9, 12, ...

Answer / उत्तर: S₁₅ = 405

Q: Find 6th term of GP: 5, 10, 20, ...

Answer / उत्तर: a₆ = 160

Q: Sum of first 3 terms of GP: 2, 6, 18

Answer / उत्तर: S₃ = 26

Q: Which term of AP: 7, 14, 21, ... is 98?

Answer / उत्तर: 14th term

Q: Find the sum of first 5 terms of AP where a = 3, d = 2

Answer / उत्तर: S₅ = 35

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Quick Recap | सार

Now you should be able to:

Recall the key ideas of this chapter without looking at the book.
Solve most NDA-style questions of this topic step by step.
Identify and avoid the most common traps used in competitive exams.

Self-check: अगर practice questions अपने आप हो रहे हैं, तो chapter strong है। जो सवाल अटकते हैं, बस उन्हें दोबारा देख लें।

Common Exam Mistakes | आम गलतियाँ

जल्दी-जल्दी में question पूरा पढ़े बिना solve करना।
Sign (+/−) या power गलत लिख देना।
Last step में calculation check न करना – आधा काम सही, answer गलत।

ध्यान रखिए: NDA में topper वो नहीं जो सब कुछ जानता है, बल्कि वो है जो छोटी-छोटी गलतियाँ avoid कर देता है।