Let's go through the problem step by step.

Problem Breakdown:

1. Initial Tree Height: 45 cm
2. Growth per Year: 10 cm

1. Finding the Amount Ashley Gets in the 9th Year:

• Height in 9th Year: The tree's height increases by 10 cm each year. Therefore, after 9 years, the height will be: $\text{Height in 9th year} = 45 + 10 \times (9-1) = 45 + 80 = 125 \text{ cm}$
• Ashley's parents give him the height of the tree in pounds as a reward. So, the amount Ashley gets in the 9th year is $125.

2. Formula for the Total Amount after $n$ Years:

• The height of the tree in the $n$th year is: $\text{Height in } n \text{th year} = 45 + 10 \times (n-1)$
• The total amount of money after $n$ years will be the sum of the tree heights for each year up to the $n$th year. This sum can be represented as: $\text{Total Amount after } n \text{ years} = \sum_{i=1}^{n} \left( 45 + 10 \times (i-1) \right)$
• Simplifying the sum: $\text{Total Amount after } n \text{ years} = n \times 45 + 10 \times \sum_{i=1}^{n} (i-1)$ The sum $\sum_{i=1}^{n} (i-1)$ is a sum of the first $n-1$ natural numbers, which is given by: $\sum_{i=1}^{n} (i-1) = \frac{(n-1)n}{2}$ So the total amount simplifies to: $\text{Total Amount after } n \text{ years} = n \times 45 + 5 \times (n-1) \times n$ $= 45n + 5n^2 - 5n = 5n^2 + 40n$ The formula for the total amount after $n$ years is: $A_n = 5n^2 + 40n$

3. Finding the Number of Years to Make a Total of $525:

• We need to solve the quadratic equation: $5n^2 + 40n = 525$ Simplifying: $n^2 + 8n - 105 = 0$ Solving this quadratic equation using the quadratic formula: $n = \frac{-8 \pm \sqrt{8^2 + 4 \times 1 \times 105}}{2 \times 1}$ $n = \frac{-8 \pm \sqrt{64 + 420}}{2} = \frac{-8 \pm \sqrt{484}}{2} = \frac{-8 \pm 22}{2}$ The two possible solutions are: $n = \frac{14}{2} = 7 \quad \text{and} \quad n = \frac{-30}{2} = -15$ Since the number of years cannot be negative, $n = 7$.

Therefore, it will take Ashley 7 years to make a total of $525.

Would you like more details or have any other questions?

Follow-up Questions:

1. What will be the total amount Ashley receives in the 12th year?
2. How much will Ashley receive in the 5th year?
3. What is the total amount Ashley receives in the first 3 years?
4. How does the amount Ashley receives change with each passing year?
5. Can you verify the quadratic formula used to find the number of years?

Tip:

For problems involving sequences, recognizing whether they are arithmetic or geometric can help simplify calculations significantly.