Question

$f(x-1)=\frac{x}{3}+f\left(x\right)$, $f\left(0\right)=2$ then value of $f\left(21\right)=?$

• A. $77$
• B. $75$
• C. $-21$
• D. $-75$

Answer 1

The correct option is D $-75$

For x = 1

$f\left(0\right)=\frac{1}{3}+f\left(1\right)...........\left(i\right)$

For x = 2

$f\left(1\right)=\frac{2}{3}+f\left(2\right)$

For x =3

$f\left(2\right)=\frac{3}{3}+f\left(3\right)..........f\left(20\right)=\frac{21}{3}+f\left(21\right)$

Similarly, putting all other equation in equation $\left(i\right)$

$f\left(0\right)=\frac{1}{3}+\frac{2}{3}+\frac{3}{3}+.........+\frac{21}{3}+f\left(21\right)$

$2=\frac{1+2+\mathrm{3....}+21}{3}+f\left(21\right)$

$2=77+f\left(21\right)$

$f\left(21\right)=-75$