To avoid paying Private Mortgage Insurance, a 20% down payment is
needed.  The house costs $98,000 and the price goes up 4% per year. 
After 8 years what is the price of the house and how much will the
down payment be?

Helpmenow --

The price of the house will be $98K * (1.04)^8 = $98K * 1.3686.

Take 20% of that number and you have the down payment.

Best regards,

Omnivorous-GA

S/price, $98.000.00, 20% D/P=19.600.00, purchase price is now, 78.400.00.
4% added for 8 years is:
Year 1. $3.136.00=$81.536.00    At 4% add on at the end of 8 years, house
     2.  3.261.44= 84.797.44    will cost $107.295.81
     3.  3.394.90= 88.189.34
     4.  3.527.57= 91.716.91
     5.  3.668.68= 95.385.59
     6.  3.815.42= 99.201.01
     7.  3.968.04=103.169.05
     8.  4.126.76=107.295.81

The problem seemed rather confusing in nature but I will try.  
In that situation you must pay the downpayment first.  Keep in mind
that the appreciation would be assumed to be compounded.
1	98,000		19600 (the amount if bought at beginning)
2	101920		
3	105996.8		
4	110236.672		
5	114646.1389		
6	119231.9844		
7	124001.2638		
8	128961.3144		25792.26287 (the amount if bought at the end)
**the problem to this question is what interest rate you get financed
at, or what your payments should be.

To buy this place with no money down and not have to pay private
mortgage insurance, split your financing into two pieces:

1.  First mortgage at 80% of the purchase price
2.  Second mortgage at 20% of the purchase price

You can even roll the closing costs into the purchase price and finance them, too.