A sequence is defined by the recursive function $f(n+1)=f(n)-2$.
If $f(1)=10$, what is $f(3)$?
The sequence is defined by the recursive relation \(f(n+1) = f(n) - 2\) with the initial term \(f(1) = 10\). A recursive definition means that each term is determined by the value of the previous term. To find a specific term like \(f(3)\), we can systematically calculate the intermediate terms of the sequence starting from the given initial term.
To find the second term \(f(2)\), we set \(n=1\) in our recursive formula:
\[f(1+1) = f(1) - 2\]
Substituting the known value of \(f(1) = 10\):
\[f(2) = 10 - 2 = 8\]
To find the third term \(f(3)\), we set \(n=2\) in our recursive formula:
\[f(2+1) = f(2) - 2\]
Substituting the calculated value of \(f(2) = 8\):
\[f(3) = 8 - 2 = 6\]
Thus, the value of the third term in the sequence is 6.
Imagine you are tracking your savings in a digital piggy bank. Every week, you decide to withdraw a fixed amount of \(\$2\) for your coffee. If you started with \(\$10\) in the first week (\(f(1)=10\)), the amount left in your bank at the start of the following weeks follows the same pattern: \(\$8\) in the second week (\(f(2)=8\)) and \(\$6\) in the third week (\(f(3)=6\)). This simple sequence helps you keep track of your remaining budget over time.