Question

If $f(x)=3^x$, prove that $f(x) + f(x+1)=4f(x)$

Answer 1

$f(x)+f(x+1) \stackrel{1}{=} 3^x+3^{x+1} \stackrel{2}{=} 3^x+3^x\cdot3^1\stackrel{3}{=}3^x+3\cdot3^x\stackrel{4}{=}4\cdot3^x\stackrel{5}{=}4\cdot f(x)$

1) Substitution from definition.

2) We use $x^{a+b}=x^a\cdot x^b$

3) We are changing the order of 3 and $3^x$.

4) We add 1 and 3 $3^x$ and we get $4\cdot 3^x$

5) Substitution from definition (but the other way than at the beginning).