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

##### 1 Answer

Bonus:

#### Explanation:

To get the next iterate, apply the function

#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

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

Subtracting the first equation from the second we get:

#6a=2#

So

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.