Question

If ${{a}_{1}},{{a}_{2}},{{a}_{3}},.....$ are positive numbers in GP then \[\log {{a}_{n}}\], \[\log {{a}_{n+1}}\] and \[\log {{a}_{n+2}}\] are in
(A) AP
(B) GP
(C) HP
(D) None of these

Answer 1

Hint: We solve this problem by first going through the concept of Geometric progression. Then we use the formula ${{b}^{2}}=ac$ if a, b, c are in GP to find the relation between ${{a}_{n}},{{a}_{n+1}}$ and ${{a}_{n+2}}$. Then we apply logarithm to the above equation and simplify it using the properties of logarithm like $\log \left( a\times b \right)=\log a+\log b$ and $\log {{a}^{n}}=n\log a$. Then we see if they follow AP or HP or GP.

Complete step-by-step answer:
First let us go through the concept of GP.
A sequence of positive numbers are said to be in Geometric progression or GP if they have a common ratio.
Three numbers a, b, c are said to be in GP if ${{b}^{2}}=ac$

We are given that ${{a}_{1}},{{a}_{2}},{{a}_{3}},.....$ are in GP.
Now let us consider the terms ${{a}_{n}},{{a}_{n+1}}$ and ${{a}_{n+2}}$. As they are in GP, we can write them using the above formula as
$a_{n+1}^{2}={{a}_{n+2}}\times {{a}_{n}}$
Now as we need the relation between \[\log {{a}_{n}}\], \[\log {{a}_{n+1}}\] and \[\log {{a}_{n+2}}\], let us apply logarithm to the above equation. By doing so we get,
$\log a_{n+1}^{2}=\log \left( {{a}_{n+2}}\times {{a}_{n}} \right)$
Now let us consider a property of logarithms,
$\log {{a}^{n}}=n\log a$
Let us also consider another property of logarithms,
$\log \left( a\times b \right)=\log a+\log b$
Using these two properties we can simplify the above equation as,
$2\log {{a}_{n+1}}=\log {{a}_{n+2}}+\log {{a}_{n}}$
We can write it as $\log {{a}_{n+1}}=\dfrac{\log {{a}_{n+2}}+\log {{a}_{n}}}{2}$
Now let us consider the property of AP,
Three numbers a, b, c are said to be in AP if they satisfy the condition,
$b=\dfrac{a+c}{2}$
So, as we can see that our above equation is in the similar form as in the definition of AP, we can say that \[\log {{a}_{n}}\], \[\log {{a}_{n+1}}\] and \[\log {{a}_{n+2}}\] are in AP.

So, the correct answer is “Option A”.

Note: The common mistake that is made by many people while solving this question is one might get confused between the AP, GP and HP. One might also make a mistake by taking the properties of logarithm wrong as $\log ab=\log a-\log b$.