Question

If f is a function such that f(0)=2, f(1)=3, f(x+2)=2f(x)-f(x+1), then f(5) is

Answer 1

Here, we have the following data:
f(0) = 2

f(1) = 3

And the functional relation is:
$\boxed{f(x+2)=2f(x)-f(x+1)}$

We can see that the equation consists of three consecutive terms, namely f(x), f(x+1) and f(x+2). Since we know two values, we can find the other values as follows:


$f(x+2) = 2f(x) - f(x+1) \\ \\ \\ \text{Put x=0} \\ \\ \\ \implies f(0+2) = 2f(0) - f(0+1) \\ \\ \\ \implies f(2) = 2f(0) - f(1) \\ \\ \\ \implies f(2) = 2(2) - 3 \\ \\ \\ \implies f(2) = 1$

Thus, we have


f(2) = 1



We again use the equation.


$f(x+2) = 2f(x) - f(x+1) \\ \\ \\ \text{Put x=1} \\ \\ \\ \implies f(1+2) = 2f(1) - f(1+1) \\ \\ \\ \implies f(3) = 2f(1) - f(2) \\ \\ \\ \implies f(3) = 2(3) - 1 \\ \\ \\ \implies f(3) = 5$


Thus, we have:

f(3) = 5



We use the equation again:

$f(x+2) = 2f(x) - f(x+1) \\ \\ \\ \text{Put x=2} \\ \\ \\ \implies f(2+2) = 2f(2) - f(2+1) \\ \\ \\ \implies f(4) = 2f(2) - f(3) \\ \\ \\ \implies f(4) = 2(1) - 5 \\ \\ \\ \implies f(4) = -3$

Thus, we have:
f(4) = -3


We use the equation one final time:

$f(x+2) = 2f(x) - f(x+1) \\ \\ \\ \text{Put x=3} \\ \\ \\ \implies f(3+2) = 2f(3) - f(3+1) \\ \\ \\ \implies f(5) = 2f(3) - f(4) \\ \\ \\ \implies f(5) = 2(5) - (-3) \\ \\ \\ \implies f(5) = 10+3 \\ \\ \\ \implies \boxed{f(5)=13}$


Thus, f(5) = 13



Hope it helps
Purva
Brainly Community