Question

The money that Ronald has in his savings account earns interest each year. He does not make any withdrawals or additional deposits. The account balance at the beginning of each year is $10,000, $10,300, $10,609, $10,927.27, and so on. Write a recursive formula and an explicit formula for the sequence.

Answer 1

A recursive formula derives the next term from the previous term, while an explicit formula derives a term from its order in the sequence.

So, how would you go about finding the recursive formula? You need to know 1 term, {u_n} and relate it to the next term {u_(n+1)}. What fact do you know about banking? That usually the amount of money in your bank account are in a geometry sequence. The next term is found by taking the previous multiplied by a specific number. Do you know what number that is? How would you find that out? Well, to look for what number to multiply things by, I would divide the numbers that are next to each other.
So 10300/10000=1.03 and 10609/10300=1.03 and so on. So you know that the next term is equal to the first term multiplied by 1.03. Can you write the recursive formula now?

For the explicit formula, start with the first time {u_1}, how did you get to the next term? multiply by 1.03. How will you get to the next term again? multiply by 1.03. So to get to the third term, you multiply the first term by 1.03 a total number of 2 times. So I would conjecture that to get to the nth term you need to multiply the first term by 1.03^(n-1). Can you write the explicit formula now?

Answer 2

nope

Answer 3

What are you stuck on?