Answer

Verified

419.1k+ views

**Hint:**In order to solve this problem you need to use the property of logarithm. That is log(ab)=loga + logb and $a\log b = \log {b^a}$ then we will be getting the equations solving those equations and knowing the properties of HP you will get the right answer.

**Complete step-by-step answer:**

It is given that, log(p+r)+log(p+r-2q)=2log(p-r)

We have to use property of logarithm log(ab)=loga + logb and $a\log b = \log {b^a}$

Now, using above identity we can write log(p+r)+log(p+r-2q)=2log(p-r) as

$\log ((p + r)(p + r - 2q)) = \log {(p - r)^2}$

On cancelling log from both sides we get the equation as,

$

\Rightarrow (p + r)(p + r - 2q) = {(p - r)^2} \\

\Rightarrow {p^2} + pr - 2qp + pr + {r^2} - 2pr \\

$

Now, ${p^2}$ and ${r^2}$ are cancelled,

Then the equation will become,

2pr – 2pq – 2qr = -2pr

4pr = 2pq + 2qr

On dividing this equation by 2pqr we get,

$ \Rightarrow \dfrac{2}{q} = \dfrac{1}{p} + \dfrac{1}{r}$

This condition shows that p, q, r makes an HP where the above condition is the harmonic mean of the HP p, q, r.

Now, we can easily see p, q, r is in HP.

**So, the correct answer is “Option C”.**

**Note:**Whenever we face such types of problems we use some important points. Like always trying to remove log from both sides of the equation by using the identity of logarithm then after some easy rearrangement we can easily identify which type of progression formed.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

10 examples of law on inertia in our daily life

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE