how mathematically did they get 0.8173 for p(3) can you break it down for me

1) Your client wants to buy a three-year zero coupon bond. You observe the following information in the market: One-year zero-coupon bond is priced at $99 per $100 face value. Two-year zero-coupon bond is priced at $95 per $100 face value. Three-year coupon bond that offers a 3% coupon rate with interest paid annually and is priced at $90 per $100 face value. What is the price of three-year zero coupon bond per $100 face value? Solution The 1-year zero price P(1) can be calculated by solving for: 100 99 = a1 +2) So, 1-year zero price is P(1) - 1 99/100 The 2-year zero price P(2) can be calculated by solving for: 95 = 100 *(1 + 2)2 So, 2-year zero price is P(2) = = 95/100 Using this 1-year and 2-year zero prices, 3-year zero price P(3) can be calculated as 3 x P(1) + 3 x P(2) + 103 x P(3) = 90 3x +3% + 103 x P(3) = 90 Solving for P(3), we get P(3) = 0.8173, thus three-year zero coupon bond price is $0.8173 * $100 = $81.73