If \(f(x)=ax+b\) and \(f(f(f(x)))=8x+21\) and if \(a,b\) are real numbers then \(a+b\) is equal to

\(2\) \(3\) \(5\) \(7\)

\(5\)

\(f(x)=ax+b\) so $f(f(x))=(a(ax+b))+b$ $=a^2x+ab+b$ $f(f(f(x)))=a^2(ax+b)+ab+b$ $=a^3x+a^2b+ab+b=8x+21$ so $a^3=8⇒a=2$ so $a^2b+ab+b=21$ $⇒4b+2b+b=21⇒b=3$ $a+b=5$