Respuesta :
Answer:
f(n)=50n+950
f(1)=1,000; f(n)=f(n-1)+50, for n≥2
Step-by-step explanation:
Sophia's grandmother gave her $1,000 to start the book fund. Sophia decided to add $50 to the fund each month from the money she earns.
Since Sophia adds a fixed amount to the college fund each month, the situation represents an arithmetic sequence with a common difference of 50.
If a is the initial value of the arithmetic sequence and d is the common difference, the explicit function below can be used to write the function.
f(n)=a+(n-1)d
Substitute a = 1,000 and d = 50 in the explicit function above and simplify.
f(n)=a+(n-1)d
f(n)=1,000+(n-1)50
f(n)=1,000+50n-50
f(n)=50n+950
So, the explicit formula that can be used to define the given situation is f(n) = 50n + 950.
Recall the recursive form of an arithmetic sequence, where d is the common difference.
f(n)=f(n-1)+d, for n≥2
For the recursive formula, the first term of the sequence, f(1), must be known. For the given situation, f(1) = 1,000.
To determine the recursive formula for the situation, define the first term, f(1) = 1,000 and substitute d = 50 into the recursive form of an arithmetic sequence as shown below.
f(1)=1,000
f(n)=f(n-1)+50, for n≥2
