Question

If three positive unequal numbers a, b, c be in H.P. then which of the following is false?

• A. $a^2+c^2>{2b}^2$
• B. $a^3+c^3>{3b}^3$
• C. ${2b}^5<a^5+b^5$
• D. $a^4+c^4>b^4$

Answer 1

The correct option is B $a^3+c^3>{3b}^3$

Given that a,b,c are in HP we know that $AM⩾GM⩾HM$

here $a,b,c$ are unequal so equality

will not hold i.e $AM>GM>HM$

$\therefore a,b,c$ are in $HP$, so $\frac{2ac}{a+c}=b$

$\begin{array}{cc}HM\text{of}a\text{and}c& =\frac{2}{\frac{1}{a}+\frac{1}{c}}\\ & =\frac{2ac}{a+c}=b\end{array}$

$G\cdot M$ of $a^{\prime }$ and $c^{\prime }=(ac{)}^{1/2}$

$=\sqrt{ac}$

Now, as $GM>HM$

so $\sqrt{ac}>b$

$\Rightarrow (\sqrt{ac}{)}^n>b^n\cdots (1),n\in N$

$\begin{array}{cc}\text{Let us take two numbers}& \text{,}{a}^n\text{and}{c}^n\\ & \cdot \frac{a^n+c^n}{2}>{(a^n\cdot c^n)}^{1/2}\\ & \Rightarrow \frac{a^n+c^n}{2}>(\sqrt{ac}{)}^n\cdots \left(2\right)\end{array}$

After analysis equations (1) and (2)

$\frac{a^n+c^n}{2}>b^n$

$\Rightarrow a^n+c^n>2b^n$

so if $n=2\to a^2+c^2>2b^2$

if $n=5\to a^5+b^5>2b^5$

if $n=4\to a^4+b^4>2b^4$

if $n=3\to a^3+b^3>2b^3$

so we can clearly say that option (B

is incorrect.