A school has decided to plant some endangered trees on $51^{st}$ 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 $10^{th}$ row ? (ii) How many more trees will be planted in the $8^{th}$ row than in the $5^{th}$ 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 $11^{th}$ row ?

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, $a_1$, 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 $n^{th}$ row, we use the formula for the $n^{th}$ term of an AP: $a_n = a_1 + (n-1)d$. Substituting $n=10$, $a_1=50$, and $d=20$, we calculate the number of trees in the $10^{th}$ row.

Step 3: Calculate difference in trees between 8th and 5th rows

First, we calculate the number of trees in the $8^{th}$ row ($a_8$) and the $5^{th}$ row ($a_5$) using the $n^{th}$ term formula. Then, we subtract the number of trees in the $5^{th}$ row from the $8^{th}$ 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: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. We set $S_n = 3200$ 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 $11^{th}$ row, we first calculate the total number of trees planted in the first 11 rows using the sum formula $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. Then, we subtract this sum from the target of 3200 trees.

More from this chapter

• find the number of terms in the AP 7 13 19......205
• is 78 atom of the AP 5 11 17 23
• which time of Ap 3,8,13,18 is 78
• the sum of the first 15 terms of an ap is 450 and its first term is 10 find its common difference
• if the (p)th term of an ap is (q) and the (q)th term is (p) then its (p+q)th term is
• if the teeth term of an ap is Q and the cute term is P then its P + Q to term is
• the fifth time of an ap is 18 and 11th term is 42 find first term
• if the sum of the first n terms of an ap is S sem is equal to 3 and square + 5 and find its common difference

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