Question

SAVINGS ACCOUNT Like Tom in Exercise 18, Sally invests $1,000 in an account that pays 5% interest, compounded annually. However, she also makes additional deposits of $50 at the end of each year. Set up and solve an initial value problem for the amount of money An in the account after n years.​

Answer 1

Answer:

if there is one year then the ans is $1050

Answer 2

Given:

Sally invests $1,000 in an account that pays 5% interest, compounded annually. However, she also makes additional deposits of $50 at the end of each year.

To Find:

Set up and solve an initial value problem for the amount of money An in the account after n years.​

Solution:

To set up a equation here we need to divide the situation in two parts first on 1000 every year the compound interest is being calculated and after 2nd year an ordinary annuity is being calculated on 50

So using the formula for compound interest an final value of ordinary annuity we can start an equation as,

$Amount=P(1+r)^n+P_{1}(\frac{(1+r)^{n-1}-1}{r})\\=1000(1+0.05)^n+50(\frac{(1+0.05)^{n-1}-1}{0.05} )\\=1000(1.05)^n+1000(\frac{1.05^n}{1.05} -1)\\=1000[1.05^n+\frac{1.05^n}{1.05}-1]\\=1000[\frac{2.05*1.05^n-1.05}{1.05} ]$

Now this equation can be used to find the amount of money in the account after n years

Hence, the equation will be$1000[\frac{2.05*1.05^n-1.05}{1.05} ]$.