Let a_{1},a_{2}, be positive real number in G.P. for each n, let A_{n},G_{n},H_{n} be respectively, the arithmetic mean, geometric mean, harmonic mean of a_{1},a_{2} a_{n}. Find an expression for the geometric mean of G_{1},G_{2}, G_{n} in terms of A_{1},A_{2}, A_{n},H_{1},H_{2},.H_{n} if a,H,H_{2},b are in H.P.

Question

Let a_1,a_2, ... be positive real number in G.P. for eachn n, let A_n,G_n,H_n be respectively, the arithmetic mean,n geometric mean, harmonic mean of a_1,a_2 ... a_n. Find ann expression for the geometric mean of G_1,G_2, ... G_n inn terms of A_1,A_2, ... A_n,H_1,H_2, ... .H_n ifn a,H,H_2,b are in H.P.

DivineJEE · Accepted Answer

Let a be the first term and r be the common ratio of the G.P. a_1, a_2, a_3 ... then a_k=a r^k-1 for k=1,2,3 ... ..nnAs a_1, a_2, ... are positive real numbersnna>0 and r>0nnCase I: When r ≠ 1nnWe have A_n=(1 / n)(a_1+a_2+ * s,+a_n)=(1 / n)(a+a r+ * s+a r^n-1)nnnn n =(1 / n)[(a(1-r^n) / 1-r)]^n=(a(1-r^n) / n(1-r)) n G_n=(a_1 a_2 ... a_n)^(1 / n) n =(a * a r * a r^2 ... a r^n-1)^(1 / n)=[a^n r^1+2+ ... (n-1)]^(1 / n) n =[a^n r^n(n-1) / 2]^(1 / n)=a r^(n-1) / 2 n and (1 / H_n)=(1 / n)((1 / a_1)+(1 / a_2)+ * s+(1 / a_n))=(1 / n)((1 / a)+(1 / a r)+ * s+frac1a r^n-1) n =(1 / n)(frac(1 / a)(1-((1 / r))^n)1-(1 / r))=(1 / n)[(1 / a)((r^n-1 / r-1)) (r / r^n)]=(1 / n)[(1-r^n / 1-r)] frac1a r^n-1 n => H_n=fracn(1-r) a r^n-11-r^nnn nnn nThus A_n H_n=(a(1-r^n) / n(1-r)) * fracn(1-r) a r^n-11-r^n==a^2 r^(n-1)=G_n^2.nnNext, let G be geometric mean of G_1, G_2, ... , G_n, thennnnn n G=(G_1 G_2 ... G_n)^1 / n => G^2 n=(G_1 G_2 ... G_n)^2 n =(A_1 A_2 ... A_n)(H_1 H_2 ... H_n)=(A_1 H_1)(A_2 H_2) ... (A_n H_n) n => G=(A_1 A_2 ... A_n H_1 H_2 ... H_n)^1 / 2 nnn nnn nCase II : When r=1nnIn this case A_n=(1 / n)(a+a+ * s+a)=annSimilarly G_n=a and H_n=annAlso A_n H_n=a^2=G^2 _nnn∴ In this case too, G=(A_1 A_2 ... A_n H_1 H_2 ... H_n)^1 / 2 nn

Explanation

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