A sequence is defined by the formula $f(n+1)=f(n)-3$. If $f(4)=22$, what is $f(1)$?
The sequence is defined by the recursive relation \(f(n+1) = f(n) - 3\). This tells us that each term in the sequence is obtained by subtracting 3 from the previous term. Such a sequence is known as an arithmetic progression, where the common difference is \(d = -3\). We are given the fourth term, \(f(4) = 22\), and our goal is to find the first term, \(f(1)\).
The general term of an arithmetic progression can be expressed using the formula \(f(n) = f(1) + (n - 1)d\), where \(d\) is the common difference. We know that \(f(4) = 22\), \(n = 4\), and \(d = -3\). Substituting these values into the formula, we get:
\[f(4) = f(1) + (4 - 1)(-3)\]
\[22 = f(1) + 3(-3)\]
\[22 = f(1) - 9\]
To find \(f(1)\), we solve the linear equation derived in the previous step by adding 9 to both sides:
\[f(1) = 22 + 9\]
\[f(1) = 31\]
Alternatively, we can work backward step by step from the known term:
\[f(4) = 22\]
\[f(3) = f(4) + 3 = 22 + 3 = 25\]
\[f(2) = f(3) + 3 = 25 + 3 = 28\]
\[f(1) = f(2) + 3 = 28 + 3 = 31\]
Both approaches consistently show that the first term is 31.
Arithmetic sequences are frequently used in financial planning, such as calculating the total amount of money in a savings account where a fixed amount is added or withdrawn periodically. For instance, if you start with an initial deposit and withdraw exactly \(\$3\) every month, you can easily project your account balance over time using the same sequence logic. This model is essential for budgeting and understanding linear growth or decline in various real-world processes.