1. **Problem statement:** A diamond necklace's selling price increases by 5% each year. The price in 2000 was 36000. (a) Write an expression for the price $n$ years later and find the price in 2008. (b) Find the total money obtained from sales over 10 years starting in 2000. 2. **Formula and explanation:** The price increases by 5% yearly, so it forms a geometric sequence with common ratio $r = 1 + 0.05 = 1.05$. The price after $n$ years is given by: $$ P_n = P_0 \times r^n $$ where $P_0 = 36000$ is the initial price. 3. **Part (a) - Expression and price in 2008:** - The year 2008 is 8 years after 2000, so $n=8$. - Expression for price after $n$ years: $$ P_n = 36000 \times 1.05^n $$ - Calculate price in 2008: $$ P_8 = 36000 \times 1.05^8 $$ Calculate $1.05^8$: $$ 1.05^8 \approx 1.477455 $$ So, $$ P_8 \approx 36000 \times 1.477455 = 53188.38 $$ 4. **Part (b) - Total money over 10 years:** - The prices form a geometric sequence from $n=0$ to $n=9$ (10 terms). - Sum of geometric series: $$ S_{10} = P_0 \times \frac{r^{10} - 1}{r - 1} $$ Calculate $r^{10}$: $$ 1.05^{10} \approx 1.628895 $$ Calculate sum: $$ S_{10} = 36000 \times \frac{1.628895 - 1}{0.05} = 36000 \times \frac{0.628895}{0.05} = 36000 \times 12.5779 = 452804.4 $$ **Final answers:** - (a) Expression: $P_n = 36000 \times 1.05^n$ - Price in 2008: approximately 53188 - (b) Total money in 10 years: approximately 452804