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If \[m\] and \[n\] are positive real numbers and \[\log m,\log \left( {\dfrac{{{m^2}}}{n}} \right),\log \left( {\dfrac{{{m^2}}}{{{n^2}}}} \right)\] are in A.P. then its general term is
A.\[\log \left( {\dfrac{{{m^r}}}{{{n^{r - 1}}}}} \right)\]
B.\[\log \left( {\dfrac{{{m^{r + 1}}}}{{{n^r}}}} \right)\]
C.\[\log {\left( {\dfrac{m}{n}} \right)^r}\]
D.\[\log \left( {\dfrac{{{m^{r - 1}}}}{{{n^{r + 1}}}}} \right)\]

Last updated date: 14th Jun 2024
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Hint: Here we will use the basics of the arithmetic progression to find the general term of the given series. Firstly we will find the value of the common difference of the series. Then we will put the value of the common difference in the general equation of the A.P. for \[{r^{th}}\]term to find the general term.

Complete step-by-step answer:
We know that the A.P. series is given as \[a,a + d,a + 2d,a + 3d,\]……. where \[d\] is the common difference and \[a\] is the first term.
First term of the series is \[\log m\].
So, \[a = \log m\]
Now we will find the common difference. Common difference is the difference between the two consecutive terms of the series. Therefore, we get
\[d = \log \left( {\dfrac{{{m^2}}}{n}} \right) - \log m\]
Now as we know the property of the logarithmic function \[\log a - \log b = \log \dfrac{a}{b}\] . So, by using this property, we get
\[ \Rightarrow d = \log \left( {\dfrac{{\dfrac{{{m^2}}}{n}}}{m}} \right)\]
Simplifying the expression, we get
\[ \Rightarrow d = \log \dfrac{m}{n}\]
We know that the \[{r^{th}}\] term of an A.P. is \[{r^{th}}{\rm{term}} = a + \left( {r - 1} \right)d\]
Now, we will find the value of the \[{r^{th}}\] term of the A.P.
Substituting the value of the first term and the common difference, we get
\[ \Rightarrow {r^{th}}{\rm{term}} = a + \left( {r - 1} \right)d = \log m + \left( {r - 1} \right)\log \dfrac{m}{n}\]
We know the property of the logarithmic function \[\log {a^b} = b\log a\] . Then by using this property , we get
\[\begin{array}{l} \Rightarrow {r^{th}}{\rm{term }} = \log m + \log {\left( {\dfrac{m}{n}} \right)^{r - 1}}\\ \Rightarrow {r^{th}}{\rm{term }} = \log m.\dfrac{{{m^{r - 1}}}}{{{n^{r - 1}}}}\end{array}\]
Simplifying the expression, we get
\[ \Rightarrow {r^{th}}{\rm{term}} = \log \dfrac{{{m^r}}}{{{n^{r - 1}}}}\]
Hence, \[\log \dfrac{{{m^r}}}{{{n^{r - 1}}}}\] is the general form of the given series.
So, option A is the correct option.
Note: Here, the terms are in AP. AP is an abbreviation for arithmetic progression and it is a sequence or series in which there is a common difference between consecutive element or numbers. The formula of arithmetic progression is used to find the number of terms in a particular series. Apart from arithmetic progression, there are two other types of progression i.e. geometric progression and harmonic progression. In geometric progression, the next number of a series and the previous number of the series has a common ratio. Harmonic progression is the reciprocal of arithmetic progression.