JEE Main Algebra — Combinatorics & Series: Sequence and Series – Chapter Test

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Difficulty: 10 medium · 10 hard

Sequence and Series MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. For positive real numbers a, b, c, the minimum value of the product (a + b + c)(1/a + 1/b + 1/c) is:

   धनात्मक वास्तविक संख्याओं a, b, c के लिए, गुणनफल (a + b + c)(1/a + 1/b + 1/c) का न्यूनतम मान है:

   • A. 1
   • B. 9
   • C. 3
   • D. 6
   Solution

   By the AM–HM inequality (or Cauchy–Schwarz), (a+b+c)(1/a+1/b+1/c) ≥ 9, with equality when a = b = c.

   समान्तर माध्य–हरात्मक माध्य असमिका (अथवा कौशी–श्वार्ज़) द्वारा, (a+b+c)(1/a+1/b+1/c) ≥ 9, तथा समानता तब होती है जब a = b = c।

2. Q2. The sum to infinity of the series Σ (from r = 1 to ∞) of 1/[r(r + 1)] is:

   श्रेणी Σ (r = 1 से ∞ तक) 1/[r(r + 1)] का अनंत तक योग है:

   • A. 3/2
   • B. 1/2
   • C. 2
   • D. 1
   Solution

   By partial fractions each term is 1/r − 1/(r+1); the sum telescopes to 1 − 1/(n+1), which tends to 1.

   आंशिक भिन्नों द्वारा प्रत्येक पद 1/r − 1/(r+1) है; योग दूरभाषित (telescope) होकर 1 − 1/(n+1) बनता है, जो 1 की ओर प्रवृत्त होता है।

3. Q3. The arithmetic mean and geometric mean of two positive numbers are 10 and 8 respectively. The harmonic mean of the same two numbers is:

   दो धनात्मक संख्याओं का समांतर माध्य और गुणोत्तर माध्य क्रमशः 10 और 8 हैं। इन्हीं दो संख्याओं का हरात्मक माध्य है:

   • A. 32/5
   • B. 25/4
   • C. 16/3
   • D. 40/7
   Solution

   For two positive numbers, GM² = AM × HM, so HM = GM²/AM = 64/10 = 32/5.

   दो धनात्मक संख्याओं के लिए, GM² = AM × HM, अतः HM = GM²/AM = 64/10 = 32/5।

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