If a variate takes value \[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},.....,a{{r}^{n-1}}\], which of the relation between means hold a) \[AH={{G}^{2}}\] b) \[\dfrac{A+H}{2}=G\] c) \[A>G>H\] d) \[A=G=H\]
ANSWER
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Hint: In this question, we are talking about arithmetic mean, geometric mean and harmonic mean. And we know that arithmetic mean, geometric mean, harmonic mean of two terms ‘a’ and ‘b’ can be represented as \[A=\dfrac{a+b}{2}\], \[G=\sqrt{ab}\] and \[H=\dfrac{2ab}{a+b}\] respectively.
Complete step by step answer: In this question, we are given a variate with terms as \[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},.....,a{{r}^{n-1}}\], and we have to find the relation between means. When we consider means of variate, we consider three means, which are arithmetic mean, geometric mean and harmonic mean. Let us consider, ‘a’ and ‘b’ be two terms, so their arithmetic mean, geometric mean and harmonic mean be \[A=\dfrac{a+b}{2}......\left( i \right)\], \[G=\sqrt{ab}......\left( ii \right)\] and \[H=\dfrac{2ab}{a+b}......\left( iii \right)\] respectively. Now, let us find out the sum of arithmetic mean and harmonic mean of any two terms, let ‘a’ and ‘b’ be the terms. Then, we can write, \[A+H\], which is equal to, \[\left( \dfrac{a+b}{2} \right)+\left( \dfrac{2ab}{a+b} \right)\] \[\Rightarrow \dfrac{{{\left( a+b \right)}^{2}}+4ab}{2\left( a+b \right)}\] \[\Rightarrow \dfrac{{{a}^{2}}+{{b}^{2}}+2ab+4ab}{2\left( a+b \right)}\] \[\Rightarrow \dfrac{{{a}^{2}}+{{b}^{2}}+6ab}{2\left( a+b \right)}\] Which is possibly not equal to geometric mean, that means, \[\dfrac{A+H}{2}\ne G\]. Hence, option (b) is incorrect. Now, let us find out the product of arithmetic mean and harmonic mean, for any two terms, let ‘a’ and ‘b’. Then, we can write it as, \[A\times H\], which is equal to \[\left( \dfrac{a+b}{2} \right)\times \left( \dfrac{2ab}{a+b} \right)\] \[\Rightarrow \dfrac{\left( a+b \right)2ab}{2\left( a+b \right)}\] \[\Rightarrow ab\] \[={{G}^{2}}\] Hence, we can see that this condition satisfies the option (a). Therefore, the correct answer is option (a).
Note: There are possibilities that one might choose option (c) or option (d) as the correct option, but it is not like that. Option (d) only satisfies when all the terms are equal and we are given a variate series. Also, option (c) is not true because the value of variate depends on the value of ‘r’ and with the change in the value of ‘r’ the result will change. So, only option (a) is correct.