Question

If $f\left(x\right)+f(x+4)=f(x+2)+f(x+6)\forall xϵRanda>0$ and f(x) is periodic, then period of f(x), is

• A. 1000
• B. 100
• C. 10000
• D. 10

Answer 1

The correct option is A 1000

$f\left(x\right)+f(x+4)=f(x+2)+f(x+6)...1\left(i\right)$.
Replacing x by x + 2 in equation (i), we get
$F(x+2)+f(x+6)=f(x+4)+f(x+8)$ . . . . .(ii)
Adding (i) and (ii) $f\left(x\right)=f(x+8)$
$f\left(5\right)=f\left(13\right)=10$
$f\left(13\right)=f\left(21\right)=\mathrm{10..}$
$f\left(805\right)=10$
100
$\therefore \sum f(5+8r)=f\left(13\right)+f\left(21\right)+...+f\left(805\right)$
r=1
=10+10+....100 times.=10 $\times$ 100=1000