# Suppose you take out a home mortgage for $160000$ at a monthly interest rate of $0.5$%. If you make payments of $1200$ per month

Suppose you take out a home mortgage for $$160000$$ at a monthly interest rate of $$0.5$$%. If you make payments of $$1200$$ per month, after how many months will the loan balance be zero. Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer.

I have thought to do the following:

As you have to pay $$0.5$$% for interest then in the first month you have to pay $$0.5$$% of $$160000=800$$ for interest and as you paid for $$1200$$, then for the house you gave $$1200-800=400$$, so $$160000-400=159600$$ was already owed and to know how much is left of the second month one does the same, takes out $$0,5$$% of $$159600=798$$ and thus paid interest $$798$$ and was due $$159198$$. This reasoning is fine? How can I generalize this and do what they ask of me in the problem? Thank you.

Let B(n) be the balance at the end of n months.
B(0) = a = 160,000.
B(n + 1) = 1.05.B(n) - 1200.

B(1) is easy to calculate. Calculate B(1) and B(2) to get a sense for B(n) and how complicated it will be. Don't use 160,000, use a to make this clearer.

All this mortgage stuff is so well known,
it's on the web, formulas and tables both.

You're doing fine so far. The next step is to try to write it in symbols instead of numbers, so that you can see the pattern easier.

Let $$P_n$$ be the amount of the loan outstanding after $$n$$ months. $$P_0$$is the amount of the loan after $$0$$ months, that is, at the beginning, so $$P_0=160000.$$ Now, if we know $$P_n,$$ how do we calculate $$P_{n+1}?$$ You already shown how to do it; take $$0.5%$$ of subtract that from $$1200$$, and the subtract the difference from $$P_n$$ $$P_{n+1}=P_n-(1200-.05P_n)=1.05P_n-1200\tag{1}$$

Now write out he same examples you did, but using symbols this time, by repeated applying equation $$(1).$$ \begin{align} P_1&=1.005P_0-1200\\ P_2&=1.005P_1-1200=1.005(1.005P_0-1200)-1200 = 1.005^2P_0-1200(1+1.005)\\ P_3&=1.005P_2-1200=1.005^3P_0-1200(1+1.005+1.005^2)\\ &\vdots \end{align}

As to what they want you to do, I'm guessing, because I don't know what tools you have available. I would think they want you to use equation $$(1)$$ to calculate the various values of $$P_n$$ perhaps using some computer spreadsheet program, and graph the values of $$P_n$$ over time, to see when the value first becomes negative, so that the loan has been paid off.

Then they want you to figure out the general formula for $$P_n,$$ a try to compute the value of $$n$$ for which $$P_n$$ = $$0$$. This will almost surely not be a whole number.