{Use of Tech} Investment problem A one-time investment of $250...

Question

{Use of Tech} Investment problem A one-time investment of \$2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰.

a. Use Newton’s method to find the value of r if the goal is to have \$3200 in the account after 5 years.

a. Use Newton’s method to find the value of r if the goal is to have $3200 in the account after 5 years.

Accepted Answer

Hello, in this problem, we are told that a money worth $$3000 is deposited in a 4-year savings account that pays a fixed annual interest rate with quarterly compounding interest. The amount in the account after 4 years is given by the function A of R equal to 3000 multiplied by 1 plus R divided by 4, raised to the power of 16. We want to find the value of R needed for the account to grow to 4000 $$after 4 years. So, we are going to use Newton's method to approximate the value of R. Newman's method is given as the following recursive formula. RN + 1 is equal to RN minus a function F of X divided by its derivative. So, what we want to do is we want to go ahead and determine the function in which the in which to approximate with and take its derivative. Then we are going to guess a starting value or sub N. So, keep in mind that the amount in the account after 4 years is given to us as the following function. A sub R is equal to 3000. Multiplied by 1 + R divided by 4, quantity rates to the power of 16. Now keep in mind that we want this equation to equal to $4000 after 4 years. If we set this equation equal to 0 by subtracting 4000, we will have 3000 multiplied by 1 plus R divided by 4, raised to the power of 16 minus 4000 equal to 0. What we have here is going to be the function F of X, or in this case F of R, that we're going to use to approximate with in Newton's method. What we are going to do now is we are also going to take the derivative of this FFR function. The derivative with the F of our function. is going to be 12,000. Multiplied by 1 plus R divided by 4, raise the power of 15. And now what we need to do is we need to go ahead and choose a value of our sub N to start the approximations with. We can choose a value between 0 and 1, so let's go ahead and let our sub N equal to 0.00. 07. So This is going to be our initial starting value, and we're going to use the functions in Newton's method. That is going to allow us to write Newton's method as R N + 1 is equal to 0.07 minus the function 3000. Multiplied by 1 plus 0.07 divided by 4, raises the power of 16 minus 4000, divided. By 12,000 Multiplied by 1 + 0.07. Divided by 4, raise to the power of 15. And with the help of a calculator, our sub N + 1 is going to approximate to B. 0.07258 Now, what we are going to do is we are going to take this new value and plug it back into Newton's method to approximate the next value. And we are going to stop when we get two values that match with each other. If we take this value of RN + 1 and plug it back into the Newton method, we get R N + 2 is approximately equal to 0.07257. And taking this value and plugging it back into Newton's method, will give us our sub N + 3, approximately equal to 0.07257. Notice that both of these values at the end match each other. And that means that we want our rate to equal to 0.07257 based off of Newton's method. Of course you want this rate to be in a percentage, so if we multiply this by 100. We will go ahead and rewrite R to be 7.257%, which is going to be the solution to this problem. So I hope this video helps you in understanding how to use Newton's method, and I will go ahead and see you all in the next video.

Key Concepts

Compound Interest Formula

Newton's Method

Derivative Calculation

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