Question

How do you find the first three iterates for the function `f(x)=-2x+5` given the initial value `x_0=2`?

Answer 1

`x_1 = 1`, `x_2 = 3` and `x_3 = -1`

Bonus: `x_n = 1/3(-2)^n+5/3`

Explanation:

To get the next iterate, apply the function `f(x)` to the previous one:

`x_1 = f(x_0) = f(2) = -2(color(blue)(2))+5 = 1`

`x_2 = f(x_1) = f(1) = -2(color(blue)(1))+5 = 3`

`x_3 = f(x_2) = f(3) = -2(color(blue)(3))+5 = -1`

Bonus - General formula for a term

To find the general formula for a term of this sequence of iterates, first note that for large `x` we have `f(x)/x ~~ -2`. So we should expect a formula like `x_n = a(-2)^n+b`.

Let's try:

`1 = x_1 = -2a+b`

`3 = x_2 = 4a+b`

Adding twice the first equation to the second, we find:

`3b=5`

So `b=5/3`

Subtracting the first equation from the second we get:

`6a=2`

So `a=1/3`

So let's try:

`x_n = 1/3(-2)^n+5/3`

Then:

`x_(n+1) = f(x_n)`

`color(white)(x_(n+1)) = -2x_n+5`

`color(white)(x_(n+1)) = -2(1/3(-2)^n+5/3)+5`

`color(white)(x_(n+1)) = 1/3(-2)^(n+1)-10/3+5`

`color(white)(x_(n+1)) = 1/3(-2)^(n+1)+5/3`

So our formula is correct.