Turn the recursive function into its equivalent explicit function:
f(1) = 18;
f(n) = f(n - 1) + 6;
n is be greater than or equal to 2*
How would I solve this? Any help is appreciated! Thanks!
*The symbol for that is the > with a line under it, I couldn't paste that in jiskha.
n ≥ 2
looks like you're starting at 18, and adding 6 to each consecutive term
An = 6 (n + 2)
Yes, n ≥ 2, thank you!
So the answer that was given to me was:
f(n) 12 + 6n; n ≥ 1
So do I just plug in the numbers for An = 6(n + 2) to get there? Sorry, I'm having a bit of trouble understanding it... I'm just unsure of how they came up with that answer.
An becomes f(n) ... distributing the 6 ... 6n + 12
f(n) = 6 n + 12
Oh wow! It was so simple I couldn't even see it! I guess I overthought it.... Thank you so much for your help!
To turn the recursive function into an equivalent explicit function, you need to express the function in a non-recursive formula that directly calculates the value of f(n) without relying on previous function values.
Let's analyze the recursive function f(n) = f(n - 1) + 6:
1. First, let's consider the base case f(1) = 18. This is important as it will serve as the starting point for building the explicit function.
2. Next, observe the pattern of the function. Each term is obtained by adding 6 to the previous term. For example:
f(2) = f(1) + 6
f(3) = f(2) + 6
f(4) = f(3) + 6
...
It is apparent that the difference between consecutive terms is always 6.
3. Now, let's create the explicit function based on the pattern identified. Since f(1) = 18, and each term is obtained by adding 6 to the previous term, we can write the formula as:
f(n) = f(1) + 6 * (n - 1)
Substituting f(1) = 18:
f(n) = 18 + 6 * (n - 1)
This explicit function calculates the value of f(n) directly without recursion.
To use the explicit function, simply input a value for n, and you can find the corresponding value of f(n).