Let's break down the problem in the image you uploaded. This question involves a sequence of tree plantings that increases by 50 trees each year, starting with 100 trees in the first year.

Part a. How many trees will they plant during the fifth year?

From the problem, we know that:

• In the 1st year, 100 trees are planted.
• In the 2nd year, 150 trees are planted.
• In the 3rd year, 200 trees are planted.
• In the 4th year, 250 trees are planted.

This follows an arithmetic sequence where the difference between the number of trees planted each year is $d = 50$. The number of trees planted in the nth year can be calculated using the formula for an arithmetic sequence:

$T_n = T_1 + (n - 1) \cdot d$

Where:

• $T_n$ is the number of trees in year $n$,
• $T_1 = 100$ is the number of trees in the first year,
• $d = 50$ is the difference between each year.

For the 5th year:

$T_5 = 100 + (5 - 1) \cdot 50 = 100 + 200 = 300$

So, 300 trees will be planted in the fifth year.

Part b. How many trees will they have planted by the end of the fifteenth year?

To find the total number of trees planted by the end of the 15th year, we need to find the sum of the trees planted over the first 15 years of the sequence. The sum $S_n$ of the first $n$ terms of an arithmetic sequence can be found using the formula:

$S_n = \frac{n}{2} \cdot (T_1 + T_n)$

Where:

• $S_n$ is the sum of trees after $n$ years,
• $T_1 = 100$ (number of trees in the first year),
• $T_n$ is the number of trees in the nth year.

First, calculate $T_{15}$:

$T_{15} = 100 + (15 - 1) \cdot 50 = 100 + 700 = 800$

Now, calculate the sum for the first 15 years:

$S_{15} = \frac{15}{2} \cdot (100 + 800) = \frac{15}{2} \cdot 900 = 7.5 \cdot 900 = 6750$

So, 6,750 trees will have been planted by the end of the fifteenth year.

Would you like any further details or explanations? Here are 5 related questions to explore further:

1. How many trees will they plant during the 10th year?
2. What is the total number of trees planted by the end of the 10th year?
3. What is the sum of trees planted over the first 20 years if the plan were extended?
4. How would the formula change if the number of trees planted increased by 100 instead of 50 each year?
5. How many trees would they have planted in total if they followed this plan for 25 years?

Tip: Arithmetic sequences are useful for predicting regular changes, such as yearly increments!