Problem 56065. Easy Sequences 74: Fibonacci Bank Account

In fantasy world where money does grow on trees, a man decided to top-up his bank account, daily. On the first day he noticed that he has initial "integer" balance of , so he made his first "integer" deposit of . On the next days, he made deposit equivalent to the day's starting balance plus the amount he deposited on the previous day. This means that on any k-th day the following formula for the amount he would deposit, holds: .
Given that on the n-th day, he made a deposit of , calculate his initial balance , his first deposit , and his final ending balance at the end of the n-th day transaction. It is also known that the first deposit he made () is the smallest possible first deposit to attain at the n-th day, using the above formula.
While the account balances can be negative, deposits are always positive (no withdrawals please).
For example, bank statements if and , are shown below. His first deposit , is the smallest possible first deposit to attain a deposit of on the 5-th day. His starting balance should be , and his ending balance at 5-th day should be .
Since this problem involves unbelievable sums of money, please reduce your answer modulo . Therefore for the above example the final answer should be:
>> k = mod([-30,20,80],2178309)
k =
2178279 20 80
HINT: Use of a big integer arithmetic implementation, is not required in this problem. You can get the modulus using Pisano Period. is the -nd Fibonacci number. Also, the ratio of the day's ending balance to the deposit approaches the Golden Ratio as the values of n gets larger.
Solve

Solution Stats

33.33% Correct | 66.67% Incorrect
Last Solution submitted on Jul 04, 2024

Problem Comments

Solution Comments

Show comments

Problem Recent Solvers2

Suggested Problems

More from this Author116

Problem Tags

easy number theory simple

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!