Chapter 5 — Arithmetic Progressions

Overview

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1. Define an arithmetic progression (AP) and state its common difference. / समांतर श्रेढ़ी (AP) को परिभाषित कीजिए और इसका सार्व अंतर बताइए।
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   An arithmetic progression is a sequence in which each term after the first is obtained by adding a fixed number, called the common difference d, to the preceding term. Thus d = a(n) - a(n-1) is constant for all terms. / समांतर श्रेढ़ी एक अनुक्रम है जिसमें पहले पद के बाद प्रत्येक पद पिछले पद में एक निश्चित संख्या, जिसे सार्व अंतर d कहते हैं, जोड़कर प्राप्त होता है। अतः d = a(n) - a(n-1) सभी पदों के लिए अचर है।

2. Write the formula for the nth term of an AP and explain its symbols. / AP के n-वें पद का सूत्र लिखिए और उसके प्रतीकों को समझाइए।
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   The nth term is a(n) = a + (n-1)d, where a is the first term, d is the common difference and n is the term number. / n-वाँ पद a(n) = a + (n-1)d है, जहाँ a पहला पद, d सार्व अंतर और n पद की संख्या है।

3. Find the 15th term of the AP: 3, 7, 11, 15, ... / AP: 3, 7, 11, 15, ... का 15वाँ पद ज्ञात कीजिए।
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   Here a = 3 and d = 4, so a(15) = 3 + (15-1)×4 = 3 + 56 = 59. / यहाँ a = 3 और d = 4, अतः a(15) = 3 + (15-1)×4 = 3 + 56 = 59।

4. Which term of the AP 21, 18, 15, ... is zero? / AP 21, 18, 15, ... का कौन-सा पद शून्य है?
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   Here a = 21, d = -3; set a + (n-1)d = 0, so 21 + (n-1)(-3) = 0, giving 3(n-1) = 21, n-1 = 7, hence n = 8. The 8th term is zero. / यहाँ a = 21, d = -3; a + (n-1)d = 0 रखें, अतः 21 + (n-1)(-3) = 0, जिससे 3(n-1) = 21, n-1 = 7, अतः n = 8। 8वाँ पद शून्य है।

5. State the formula for the sum of the first n terms of an AP. / AP के प्रथम n पदों के योग का सूत्र लिखिए।
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   The sum is S(n) = (n/2)[2a + (n-1)d], which can also be written as S(n) = (n/2)(a + l), where l is the last term. / योग S(n) = (n/2)[2a + (n-1)d] है, जिसे S(n) = (n/2)(a + l) भी लिखा जा सकता है, जहाँ l अंतिम पद है।

6. Find the sum of the first 20 terms of the AP: 5, 8, 11, ... / AP: 5, 8, 11, ... के प्रथम 20 पदों का योग ज्ञात कीजिए।
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   Here a = 5, d = 3, n = 20; S(20) = (20/2)[2×5 + (20-1)×3] = 10[10 + 57] = 10×67 = 670. / यहाँ a = 5, d = 3, n = 20; S(20) = (20/2)[2×5 + (20-1)×3] = 10[10 + 57] = 10×67 = 670।

7. How can you check whether three numbers a, b, c are in AP? / कैसे जाँचें कि तीन संख्याएँ a, b, c समांतर श्रेढ़ी में हैं?
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   Three numbers a, b, c are in AP if and only if the differences are equal, i.e. b - a = c - b, which is the same as 2b = a + c (b is the arithmetic mean of a and c). / तीन संख्याएँ a, b, c समांतर श्रेढ़ी में हैं यदि और केवल यदि अंतर बराबर हों, अर्थात b - a = c - b, जो 2b = a + c के समान है (b, a और c का समांतर माध्य है)।

8. Find the sum of the first 30 positive integers divisible by 6. / 6 से विभाज्य प्रथम 30 धनात्मक पूर्णांकों का योग ज्ञात कीजिए।
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   The numbers form an AP 6, 12, 18, ... with a = 6, d = 6, n = 30; S(30) = (30/2)[2×6 + (30-1)×6] = 15[12 + 174] = 15×186 = 2790. / संख्याएँ AP 6, 12, 18, ... बनाती हैं जिसमें a = 6, d = 6, n = 30; S(30) = (30/2)[2×6 + (30-1)×6] = 15[12 + 174] = 15×186 = 2790।

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