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).