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Explain this and how to do this in general, but also this question specifically.

Explain this and how to do this in general, but also this question specifically.-example-1
User Tyshawn
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The function f(n) = 2/3 * f(n - 1) + 1 is linear, as it can be expressed in the form f(n) = (2/3)^n * f(1) + ((2/3)^n - 1), demonstrating linearity in f(1). Graphically, the plot of f(n) against n forms a straight line.

Yes, the function f(n) = 2/3 * f(n - 1) + 1 is linear. A linear function is a function that can be expressed in the form f(x) = mx + b, where m and b are constants.

To see that f(n) = 2/3 * f(n - 1) + 1 is linear, we can rewrite it as follows:

f(n) = (2/3) * f(n - 1) + 1

f(n) = (2/3) * (f(n - 2) + 1) + 1

f(n) = (2/3)^2 * f(n - 2) + (2/3) + 1

...

In general, we can write f(n) as follows:

f(n) = (2/3)^n * f(1) + ((2/3)^n - 1)

We can see that this function is linear in f(1), which is a constant. Therefore, f(n) is a linear function.

Another way to see that f(n) is linear is to plot it on a graph. If we plot f(n) against n, we will get a straight line.

User Charas
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