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If n is a root of `x^(2)(1-ac)-x(a^(2)+c^(2))-(1+ac)=0` and if n harmonic means are inserted between a and c, find the difference between the first and the last means. |
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Answer» Let `H_(1),H_(2),H_(3),"....",H_(n),` are n harmonic means, then `a,H_(1),H_(2),H_(3),"....",H_(n),b` are in HP. `:.(1)/(a),(1)/(H_(1)),(1)/(H_(2)),(1)/(H_(3)),"....",(1)/(H_(n)),(1)/(b)` are in AP. If d be the common difference, then `(1)/(c )=(1)/(a)+(n+2-1)d` `:. d=((a-c))/(ac(n+1))" " "......(i)"` `implies (1)/(h_(1))=(1)/(a)+d` and `(1)/(h_(n))=(1)/(c)-d` `:.h_(1)-h_(n)=(a)/(1+ad)-(c)/(1-cd)=(a)/(1+(a-c)/(a(n+1)))-(c)/(1-(a-c)/(a(n+1)))` `=(ac(n+1))/(an+a)-(ac(n+1))/(an+c)=ac(n+1)((1)/(cn+a)-(1)/(an+c))` `=ac(n+1)((an+c-cn-a)/(acn^(2)+(a^(2)+c^(2))n+ac))` `=(ac(a-c)(n^(2)-1))/(acn^(2)+(a^(2)+c^(2))n+ac)" " "....(ii)"` But given n is a root of `x^(2)(1-ac)-x(a^(2)+c^(2))-(1+ac)=0` Then, `n^(2)(1-ac)-n(a^(2)+c^(2))-(1+ac)=0` `acn^(2)+(a^(2)+c^(2))n+ac=n^(2)-1` then from Eq. (ii), `h_(1)-h_(1)=(ac+(a-c)(n^(2)-1))/((n^(2)-1))=ac(a-c)` |
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