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Ex 9.3, 27 - Find value of n so that an+1 + bn+1 / an + bn - Ex 9.3

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
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Ex9.3, 27 Find the value of n so that (๐‘Ž^(๐‘› + 1) +๐‘^(๐‘› + 1))/(๐‘Ž^(๐‘› ) +๐‘^๐‘› ) may be the geometric mean between a and b. We know that geometric mean between a & b is a & b = โˆšab It is given that G.M. between a & b = (๐‘Ž^(๐‘› + 1) +๐‘^(๐‘› + 1))/(๐‘Ž^(๐‘› ) +๐‘^(๐‘› ) ) โˆšab = (๐‘Ž^(๐‘› + 1) +๐‘^(๐‘› + 1))/(๐‘Ž^(๐‘› ) +๐‘^(๐‘› ) ) ใ€–"(ab)" ใ€—^(1/2) = (๐‘Ž^(๐‘› + 1) +๐‘^(๐‘› + 1))/(๐‘Ž^(๐‘› ) +๐‘^(๐‘› ) ) ใ€–"(ab)" ใ€—^(1/2) (an +bn) = an + 1 + bn + 1 ใ€–"a" ใ€—^(1/2) ๐‘^(1/2) (an +bn) = an + 1 + bn + 1 ใ€–"a" ใ€—^(1/2) an ๐‘^(1/2) + ใ€–"a" ใ€—^(1/2) bn ๐‘^(1/2) = an + 1 + bn + 1 ๐‘Ž^(1/2 + ๐‘› ) ๐‘^(1/2) + ใ€–"a" ใ€—^(1/2) ๐‘^(1/2 + ๐‘› )= an + 1 + bn + 1 ๐‘Ž^(1/2 + ๐‘› ) ๐‘^(1/2) โ€“ an + 1 = bn + 1 โ€“ ใ€–"a" ใ€—^(1/2) ๐‘^(1/2 + ๐‘› ) ๐‘Ž^(1/2 + ๐‘› ) ๐‘^(1/2) โ€“ ๐‘Ž^(๐‘› + 1/2 + 1/2) = ๐‘^(๐‘› + 1/2 + 1/2) โ€“ ๐‘Ž^(1/2) ๐‘^(1/2 + ๐‘› ) ๐‘Ž^(1/2 + ๐‘› ) [๐‘^(1/2) โ€“ ๐‘Ž^(1/2)] = ๐‘^(๐‘› + 1/2 ) [๐‘^(1/2) โ€“ ๐‘Ž^(1/2)] ๐‘Ž^(1/2 + ๐‘› )= ๐‘^(๐‘› + (1 )/2 "[" ๐‘^(1/2) " โˆ’ " ๐‘Ž^(1/2) "] " )/(๐‘^(1/2) " โˆ’ " ๐‘Ž^(1/2) ) ๐‘Ž^(1/2 + ๐‘› )= ๐‘^(๐‘› +1/2) (๐‘Ž/๐‘)^(1/2 + ๐‘›) = 1 (๐‘Ž/๐‘)^(1/2 + ๐‘›)= (๐‘Ž/๐‘)^0 Comparing powers 1/2 + n = 0 n = โ€“ 1/2 Hence value of n is - 1/2

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