A school has decided to plant some endangered trees on 51st World Environment Day in the nearest park. They have decided to plant those trees in few concentric circular rows such that each succeeding row has 20 more trees than the previous one. The first circular row has 50 trees.
Based on the above given information, answer the following questions :
(i) How many trees will be planted in the 10th row ?
(ii) How many more trees will be planted in the 8th row than in the 5th row ?
(iii) (a) If 3200 trees are to be planted in the park, then how many rows are required ?
OR
(b) If 3200 trees are to be planted in the park, then how many trees are still left to be planted after the 11th row ?
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Step-by-Step Solution
Step 1: Identify the arithmetic progression
The problem describes a scenario where the number of trees in each succeeding row increases by a constant amount. This indicates an arithmetic progression (AP). The first term, a1, is the number of trees in the first row, which is 50. The common difference, d, is the constant increase in the number of trees, which is 20.
Step 2: Calculate trees in the 10th row
To find the number of trees in the nth row, we use the formula for the nth term of an AP: an=a1+(n−1)d. Substituting n=10, a1=50, and d=20, we calculate the number of trees in the 10th row.
Step 3: Calculate difference in trees between 8th and 5th rows
First, we calculate the number of trees in the 8th row (a8) and the 5th row (a5) using the nth term formula. Then, we subtract the number of trees in the 5th row from the 8th row to find the difference.
Step 4: Calculate the number of rows for 3200 trees (Part iii a)
To find the number of rows required to plant 3200 trees, we use the formula for the sum of an AP: Sn=2n[2a1+(n−1)d]. We set Sn=3200 and solve the resulting quadratic equation for n. Since the number of rows cannot be negative, we take the positive solution.
Step 5: Calculate trees left after 11th row (Part iii b)
To find how many trees are left after the 11th row, we first calculate the total number of trees planted in the first 11 rows using the sum formula Sn=2n[2a1+(n−1)d]. Then, we subtract this sum from the target of 3200 trees.