Question

If $a+b+c+d=63,$ where $a,b,c,d\in I^{+},$ then the maximum value of $ab+bc+cd$ is

• A. $990$
• B. $991$
• C. $992$
• D. $994$

Answer 1

The correct option is B $991$

$ab+bc+cd=ab+bc+cd+ad-ad$
$\Rightarrow ab+bc+cd=(a+c)(b+d)-ad$
Here, $ad$ should be minimized, $\Rightarrow a=d=1$
$\Rightarrow ab+bc+cd=(1+c)(1+b)-1$
Also $a+b+c+d=63\Rightarrow b+c=61$
To maximization, $b=31\text{and}c=30$
$ab+bc+cd=(1+30)(1+31)-1=991$
$\therefore max(ab+bc+cd)=991$