Answer

Verified

393k+ views

**Hint:**Here, we need to prove that \[\log a\], \[\log b\], and \[\log c\] are in A.P. We will use the formula for \[{n^{{\rm{th}}}}\] term of a G.P. to form an equation. Then, we will take the logarithm on both the sides. Finally, we will use the rules of logarithms and simplify the equation to prove that \[\log a\], \[\log b\], and \[\log c\] are in A.P. Any three numbers \[x\], \[y\], and \[z\] are in A.P. if \[x + z = 2y\].

**Formula Used:**

We will use the following formulas:

1.The \[{n^{{\rm{th}}}}\] term of a G.P. is given by the formula \[{a_n} = a{r^{n - 1}}\], where \[a\] is the first term and \[r\] is the common ratio.

2.\[\log {x^n} = n\log x\].

3.\[\log \left( {xy} \right) = \log x + \log y\].

**Complete step-by-step answer:**First, we will use the formula for the \[{n^{{\rm{th}}}}\] term of a G.P.

It is given that \[a\], \[b\], and \[c\] are in G.P.

Therefore, the first term of the G.P. is \[a\], \[b\] is the second term of the G.P., and \[c\] is the third term of the G.P.

Substituting \[n = 2\] in the formula for \[{n^{{\rm{th}}}}\] term of a G.P., \[{a_n} = a{r^{n - 1}}\], we get

\[ \Rightarrow {a_2} = a{r^{2 - 1}}\]

Simplifying the expression, we get

\[\begin{array}{l} \Rightarrow {a_2} = a{r^1}\\ \Rightarrow {a_2} = ar\\ \Rightarrow b = ar\end{array}\]

Substituting \[n = 3\] in the formula for \[{n^{{\rm{th}}}}\] term of a G.P., \[{a_n} = a{r^{n - 1}}\], we get

\[ \Rightarrow {a_3} = a{r^{3 - 1}}\]

Simplifying the expression, we get

\[\begin{array}{l} \Rightarrow {a_3} = a{r^2}\\ \Rightarrow c = a{r^2}\end{array}\]

Multiplying both sides by the first term \[a\], we get

\[\begin{array}{l} \Rightarrow c \times a = a{r^2} \times a\\ \Rightarrow ac = {a^2}{r^2}\end{array}\]

Rewriting the expression, we get

\[ \Rightarrow ac = {\left( {ar} \right)^2}\]

Substituting \[b = ar\] in the equation, we get

\[ \Rightarrow ac = {b^2}\]

Now, we will prove that \[\log a\], \[\log b\], and \[\log c\] are in A.P.

We know that if \[x = y\], then \[\log x = \log y\].

Therefore, since \[ac = {b^2}\], we get

\[ \Rightarrow \log \left( {ac} \right) = \log \left( {{b^2}} \right)\]

The logarithm of a number raised to a power can be written as the product of the power, and the logarithm of the number. This can be written as \[\log {x^n} = n\log x\].

Substituting \[x = b\] and \[n = 2\] in the rule of logarithms \[\log {x^n} = n\log x\], we get

\[ \Rightarrow \log {b^2} = 2\log b\]

Substituting \[\log {b^2} = 2\log b\] in the equation \[\log \left( {ac} \right) = \log \left( {{b^2}} \right)\], we get

\[ \Rightarrow \log \left( {ac} \right) = 2\log b\]

The logarithm of the product of two numbers can be written as the sum of the logarithms of the two numbers. This can be written as \[\log \left( {xy} \right) = \log x + \log y\]. Here, the base of the terms need to be equal.

Substituting \[x = a\] and \[y = c\] in the rule of logarithms \[\log \left( {xy} \right) = \log x + \log y\], we get

\[ \Rightarrow \log \left( {ac} \right) = \log a + \log c\]

Substituting \[\log \left( {ac} \right) = \log a + \log c\] in the equation \[\log \left( {ac} \right) = 2\log b\], we get

\[ \Rightarrow \log a + \log c = 2\log b\]

We know that three numbers \[x\], \[y\], and \[z\] are in A.P. if \[x + z = 2y\].

Therefore, since \[\log a + \log c = 2\log b\], the numbers \[\log a\], \[\log b\], and \[\log c\] are in A.P.

**Hence, we have proved that \[\log a\], \[\log b\], and \[\log c\] are in A.P.**

**Note:**An arithmetic progression is a series of numbers in which each successive number is the sum of the previous number and a fixed difference. The fixed difference is called the common difference.

A geometric progression is a series of numbers in which each successive number is the product of the previous number and a fixed ratio. The fixed ratio is called the common ratio.

Here, we need to keep in mind different rules of logarithms to simplify the equation. If we don’t know the rules we might make a mistake by writing \[\log \left( {xy} \right) = \log x - \log y\] instead of \[\log \left( {xy} \right) = \log x + \log y\], which will incur the wrong answer.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Name 10 Living and Non living things class 9 biology CBSE

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

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

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE