Question

If $a,b,c,dϵR^{+}$ and a, b, c, d are in H.P, then

• A. a+d > b+c
• B. a+b > c+d
• C. a+c > b+d
• D. none of these

Answer 1

The correct option is A a+d > b+c

AM between a and c is $\frac{1}{2}(a+c)$

and $HM$ between a and c is b.

$AM>HM$

$\frac{a+c}{2}>b\Rightarrow a+c>2b$ $\left(i\right)$

Similarly, $C$ is $HM$. between $b$ and $d$

And AM between $b$ and $d$ is $\frac{b+d}{2}$

$AM>HM$

$\frac{b+d}{2}>c\Rightarrow b+d>2c$ $\left(ii\right)$

add (i) and (ii),

$a+b+c+d>2(b+c)$

$\Rightarrow a+d>b+c$

$\therefore$ option $A$ is correct.