{Use of Tech} A savings plan
James begins a savings plan...

Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits $100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits $100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = $0.

b.Find a recurrence relation that generates the sequence {Bₙ}.

Accepted Answer

Below there. Today we want to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A retirement account receives $200 at the start of each month and earns 0.6% interest monthly. Let R subscript N be the balance after the Nth deposit with capital R subscript zero is equal to $0. What is the reoccurrence relation for R subscript N? Awesome. So it appears for this particular prompt, we're ultimately trying to determine what is the reoccurrence relation for capital R subscript N, and to keep things simple, I'll just call it RN for short. So we're trying to figure out what this reoccurrence relation is. So another way of saying this is we're trying to figure out the expression or the equation for our subscript N. So we're trying to figure out what our subscript N or this RN is equal to. So now that we know what we're ultimately trying to solve for, our first step is we need to understand the process that is occurring in this particular prompt. So we need to recall a note that based on what we're told in this prom, that every month the interest is added first, then the $200 is deposited. So that means that the account earns 0.6% interest monthly, and this value RN is the balance after the end deposit, and R0, our subscript zero is equal to $0 because that is the initial balance, which I'll call IB for short. Awesome. So once again, the initial balance, so that's the beginning balance will be $0. So now we need to write our expression for, or rather, we can begin to start to write the equation or the expression for the reoccurrence relation. So let us suppose that after the parenthesis N minus 1 if deposit, the balance is or rather subscript N minus 1. So that will mean that at the start of the nth month, our first condition will be the interest is added first, so our subscript N minus 1 plus 0.006 multiplied by our subscript N minus 1 will be equal to 1.006 multiplied by R subscript N minus 1. Fantastic. And our second step is that the deposit of $200 is added after the fact, so that will mean we can go ahead. So I'll make a note that this is when the $200 is added. So we will take 1.006 multiplied by our subscript N minus 1 plus $200. So that will mean that after the end deposit and interest, our subscript N will be equal to 1.006 multiplied by our subscript N minus 1 plus $200. And that's it, we've officially solved for this problem. Hooray, we did it. So once again, our final answer for the reoccurrence relation will have to be as follows. Capital R subscript N is equal to 1.006 multiplied by R subscript N minus 1 plus $200. And that's it. That's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Recurrence Relations

Compound Interest

Sequence and Series in Financial Contexts

Watch next