Question

Martina opens a savings account with an initial deposit and makes no other deposits or withdrawals. She earns interest on her initial deposit. The total amount of money in her savings account at the end of each year is represented by the sequence shown.
100, 105, 110.25, ...
Which recursive formula can be used to determine the total amount of money earned in any year based on the amount earned in the previous year?

A. f(n + 1) = f(n) + 5
B. f(n + 1) = 5f(n)
C. f(n + 1) = 1.05f(n)
D. f(n + 1) = 0.05f(n)

Answer 1

1.05 should be it

Step-by-step explanation:

In a recursive formula, each term of the sequence is found by using the previous term.  In this case, the sequence is geometric because we are multiplying each term by the same constant, called a common ratio.

To find this common ratio, we find the percent of change between terms.  We do this by using the formula

% change = (amount of change)/(original amount)

The amount of change between the first two terms is 105-100 = 5.  This makes the percent change

5/100 = 0.05

The amount of change between the second two terms is 110.25-105 = 5.25.  This makes the percent change

5.25/105 = 0.05

However, the common ratio in the sequence is not 0.05.  This is because multiplying by this decimal would make the terms decrease instead of increasing.  In this case, we are taking 100% of the number and adding an extra 5% (0.05 = 5%); this makes our common ratio 105% = 1.05.

This gives us the recursive formula