Sequences And Series Ques 22
- If $a, b, c$ are positive real numbers, then prove that
$ \{(1+a)(1+b)(1+c)\}^7>7^7 a^4 b^4 c^4 $
(2004, 4M)
Show Answer
Solution: Here, $(1+a)(1+b)(1+c)$
$ =1+a+b+c+a b+b c+c a+a b c $ $\quad$ ……..(i)
Since, $\frac{a+b+c+a b+b c+c a+a b c}{7} \geq\left(a^4 b^4 c^4\right)^{1 / 7} \quad$ [using AM $\geq$ GM]
$\Rightarrow \quad a+b+c+a b+b c+c a+a b c \geq 7\left(a^4 b^4 c^4\right)^{1 / 7}$
$\Rightarrow \quad 1+a+b+c+c a+a b c>7\left(a^4 b^4 c^4\right)^{1 / 7}$ $\quad$ ……..(ii)
From Eqs. (i) and (ii), we get
$ (1+a)(1+b)(1+c)>7\left(a^4 b^4 c^4\right)^{1 / 7} $
or $\{(1+a)(1+b)(1+c)\}^7>7^7\left(a^4 b^4 c^4\right)$
BETA
