Question

We are giving the concept of A.M of $m^{th}$ power.Let $a,b>0$ and $a\ne b$ and let $m$ be a real number. Then
$\frac{a^m+b^m}{2}>{\left(\frac{a+b}{2}\right)}^m$ if $m\in R-[0,1]$
However, if $m\in (0,1)$ then $\frac{a^m+b^m}{2}<{\left(\frac{a+b}{2}\right)}^m$.
Obviously, if $m\in \{0,1\}$ then $\frac{a^m+b^m}{2}={\left(\frac{a+b}{2}\right)}^m$.
On the basis of the above information, answer the following questions:

The correct statement for sequence $C$ is:

• A. First successive differences form an A.P with common difference $\ln 4$
• B. First successive differences form an G.P with common ratio $4$
• C. Second successive differences form an A.P with common difference $\ln 2$
• D. Second successive differences form an G.P with common ratio $2$

Answer 1

The correct option is A First successive differences form an A.P with common difference $\ln 4$

Given$:$ Sequence $C:\ln 2,\ln 4,\ln 32,\ln 1024,...$
or $C:\ln 2,\ln 2^2,\ln 2^5,\ln 2^{10},...$
or $C:\ln 2,2\ln 2,5\ln 2,10\ln 2,...$
First Successive difference of $C:$ is $2\ln 2-\ln 2,5\ln 2-2\ln 2,10\ln 2-5\ln 2,...$
or $\ln 2,3\ln 2,5\ln 2,..$ where first term is $\ln 2$ and common difference is $3\ln 2-\ln 2=2\ln 2=\ln 2^2=\ln 4$
Thus the first successive difference of sequence $C$ is in A.P with common difference $\ln 4$