SOLUTION: you started a job and got paid at the end of the month and decided to put 1000.00 peso from each paycheck into special account. suppose that account is earning annual interest rate

Question 1184014: you started a job and got paid at the end of the month and decided to put 1000.00 peso from each paycheck into special account. suppose that account is earning annual interest rate of 3% compounded monthly, how much will be in the account at the end of a year?
1. find the nth term of your earnings.
2. how much will the 1000.00 peso deposited at the end of 10th month and its interest be worth at the end of the year?
3.what do you notice about the sequence?
4. describe how to calculate the amount of the money you will haveat the end of 12 months.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
answers to your questions are below:

you started a job and got paid at the end of the month and decided to put 1000 peso from each paycheck into special account.

suppose that account is earning annual interest rate of 3% compounded monthly. how much will be in the account at the end of a year?

interest rate is equal to .03 per year / 12 = .0025 per month
interest rate growth factor is equal to 1 plus that = 1.0025 per month.

the amount that will be in the account at the end of a year will be equal to 1000 * (1 - 1.0025 ^ 12) / (1 - 1.0025) = 12166.38277. ***** 1

1. find the nth term of your earnings.

the nth term of your savings is given by the formula An = 1000 * r ^ (n - 1)

An is the value of your savings in the nth term since you invested it.
r is the interest rate growth factor.
n is the term number.

for example:

you invest 1000 today and want to know it's value 5 months from now.
the value will be using the An formula.
it becomes:
A5 = 1000 * 1.0025 ^ (5 - 1) = 1000 * 1.0025 ^ 4 = 1010.037563.

2. how much will the 1000.00 peso deposited at the end of 10th month and its interest be worth at the end of the year?

1000 deposited at the end of the 10th month will be equal to 1000 * 1.0025 ^ 2 = 1005.00625.

the 1000 is equal to 1000 at the end of the 10th month.
at the end of the 12th month, it will have earned 2 months interest which is equal to 1.0025 ^ 2.
1000 * 1.0025 ^ 2 = 1005.00625.

3.what do you notice about the sequence?

it looks very much like a geometric sequence.

4. describe how to calculate the amount of the money you will have at the end of 12 months.

the formula to use here is Sn = (1 - r ^ n) / (1 - r)
Sn is the sum of the terms from first term to nth term.
r is the interest rate growth factor per month.
n is the number of terms.

using this formula, you get:

S12 = 1000 * (1 - 1.0025 ^ 12) / (1 - 1.0025) = 12166.38277.

it's the same formula used above when we calculated that the amount of money in the account at the end of 12 months is equal to 12166.38277.
see ***** 1

bottom line:

you are using geometric sequence formula to solve this problem.

An = 1000 * 1.0025 ^ (n - 1) to find the value of an investment in the nth term from when it was invested.

Sn = 1000 * (1 - 1.0025 ^ n) / (1 - 1.0025) to find the sum of the terms from 1 to n.

let me know if you have any questions.

reference on geometric sequence formula can be found here:

https://www.mathsisfun.com/algebra/sequences-sums-geometric.html

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