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Ex 9.2, 15 - If an + bn / an-1 + bn-1 is AM between a, b - Ex 9.2

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
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Ex 9.2 , 15 If (๐‘Ž^๐‘› + ๐‘^๐‘›)/(๐‘Ž^(๐‘›โˆ’1) + ๐‘^(๐‘›โˆ’1) ) is the A.M. between a and b, then find the value of n. We know that arithmetic mean between a & b is A.M. = (a + b)/2 It is given that AM between a & b is (๐‘Ž^๐‘› + ๐‘^๐‘›)/(๐‘Ž^(๐‘›โˆ’1) + ๐‘^(๐‘›โˆ’1) ) So, (๐‘Ž^๐‘› + ๐‘^๐‘›)/(๐‘Ž^(๐‘›โˆ’1) + ๐‘^(๐‘›โˆ’1) ) = (a + b)/2 2(an + bn) = (a + b) (an โ€“ 1 + bn โ€“ 1) 2an + 2bn = a(an โ€“ 1 + bn โ€“ 1) + b(an โ€“ 1 + bn โ€“ 1) 2an + 2bn = aan โ€“ 1 + abn โ€“ 1 + ban โ€“ 1 + bbn โ€“ 1 2an + 2bn = a1 . an โ€“ 1 + abn โ€“ 1 + ban โ€“ 1 + b1 . bn โ€“ 1 2an + 2bn = a1 + n โ€“ 1 + abn โ€“ 1 + ban โ€“ 1 + b1 + n โ€“ 1 2an + 2bn = a1 + n โ€“ 1 + abn โ€“ 1 + ban โ€“ 1 + b1 + n โ€“ 1 2an + 2bn = an + abn โ€“ 1 + ban โ€“ 1 + bn 2an + 2bn โ€“ an โ€“ abn โ€“ 1 โ€“ an โ€“ 1 b โ€“ bn = 0 2an โ€“ an + 2bn โ€“ bn - abn โ€“ 1 โ€“ an - 1 b = 0 an + bn โ€“ abn โ€“ 1 โ€“ an โ€“ 1 b = 0 an โ€“ an โ€“ 1 b + bn โ€“ a bn โ€“ 1 = 0 a.an โ€“ 1 โ€“ an โ€“ 1 b + b.bn โ€“ 1 โ€“ a bn โ€“ 1 = 0 an โ€“ 1 (a โ€“ b) โ€“ bn โ€“ 1 (a โ€“ b) = 0 (an โ€“ 1 โ€“ bn โ€“ 1)(a โ€“ b) = 0 โˆด an โ€“ 1 โ€“ bn โ€“ 1 = 0 Solving an โ€“ 1 = bn โ€“ 1 an โ€“ 1 = bn โ€“ 1 ๐‘Ž^(๐‘› โˆ’1)/(๐‘^(๐‘› โˆ’1) ) = 1 (๐‘Ž/๐‘)^(๐‘› โˆ’1) = 1 (๐‘Ž/๐‘)^(๐‘› โˆ’1) = (๐‘Ž/๐‘)^0 Comparing powers n โ€“ 1 = 0 n = 1 Hence n = 1

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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