A sequence is defined recursively by the formula $f(n+1)=f(n)+3$. The first term of the sequence is -4. What is the next term in the sequence?
The problem provides a sequence defined by the recursive formula \(f(n+1) = f(n) + 3\) and identifies the first term as \(f(1) = -4\). A recursive formula allows us to find the next term of a sequence if we know the current term. To find the next term, which is \(f(2)\), we need to set \(n=1\) in the given formula.
To find the second term \(f(2)\), we use the recurrence relation with \(n = 1\):
\[f(1+1) = f(1) + 3\]
\[f(2) = f(1) + 3\]
Now, substitute the known value of the first term, \(f(1) = -4\), into this equation:
\[f(2) = -4 + 3\]
Performing the addition, we find:
\[f(2) = -1\]
The calculation shows that the second term of the sequence is \(-1\). This matches the second option provided in the problem description. Thus, the correct value for the next term in the sequence is \(-1\). The final answer is \(\boxed{-1}\).
Recursive sequences are often used to model growth or change over equal intervals of time. For example, if a savings account starts with a balance of \(-4\) (representing a debt) and you pay off \(\$3\) of interest or add \(\$3\) to the principal each month, you can use a recursive sequence to track the balance. Understanding how these sequences progress is essential for financial planning and predicting future outcomes in various real-world scenarios.