Last updated at May 29, 2018 by Teachoo
Transcript
Ex9.3, 27 Find the value of n so that (๐^(๐ + 1) +๐^(๐ + 1))/(๐^(๐ ) +๐^๐ ) may be the geometric mean between a and b. We know that geometric mean between a & b is a & b = โab It is given that G.M. between a & b = (๐^(๐ + 1) +๐^(๐ + 1))/(๐^(๐ ) +๐^(๐ ) ) โab = (๐^(๐ + 1) +๐^(๐ + 1))/(๐^(๐ ) +๐^(๐ ) ) ใ"(ab)" ใ^(1/2) = (๐^(๐ + 1) +๐^(๐ + 1))/(๐^(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^(1/2 + ๐ ) ๐^(1/2) + ใ"a" ใ^(1/2) ๐^(1/2 + ๐ )= an + 1 + bn + 1 ๐^(1/2 + ๐ ) ๐^(1/2) โ an + 1 = bn + 1 โ ใ"a" ใ^(1/2) ๐^(1/2 + ๐ ) ๐^(1/2 + ๐ ) ๐^(1/2) โ ๐^(๐ + 1/2 + 1/2) = ๐^(๐ + 1/2 + 1/2) โ ๐^(1/2) ๐^(1/2 + ๐ ) ๐^(1/2 + ๐ ) [๐^(1/2) โ ๐^(1/2)] = ๐^(๐ + 1/2 ) [๐^(1/2) โ ๐^(1/2)] ๐^(1/2 + ๐ )= ๐^(๐ + (1 )/2 "[" ๐^(1/2) " โ " ๐^(1/2) "] " )/(๐^(1/2) " โ " ๐^(1/2) ) ๐^(1/2 + ๐ )= ๐^(๐ +1/2) (๐/๐)^(1/2 + ๐) = 1 (๐/๐)^(1/2 + ๐)= (๐/๐)^0 Comparing powers 1/2 + n = 0 n = โ 1/2 Hence value of n is - 1/2
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