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प्रश्न
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Sanjana being a plant lover decides to convert her balcony into a beautiful garden full of plants. She bought few plants with pots for her balcony. She placed the pots in such a way that the number of pots in the first row is 2, second row is 5, third row is 8, and so on. |
Based on the given information, answer the following questions.
- Find the number of pots placed in the 10th row.
- Find the difference in the number of pots placed in the 5th row and the 2nd row.
- If Sanjana wants to place 100 pots in total then find the total number of rows formed in the arrangement.
- If Sanjana has sufficient space for 12 rows then how many total number of pots are placed by her with the same arrangement?
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उत्तर
From the given situation, the number of pots arranged in successive rows are: \[2, 5, 8, \ldots\]
This sequence forms an Arithmetic Progression (AP) where:
First term, \[a = 2\]
Common difference, \[d = 5 - 2 = 3\]
i. Number of pots placed in the 10th row:
Formula: \[T_n = a + (n - 1)d\]
Substitution: \[T_{10} = 2 + (10 - 1) \times 3\]
Calculation: \[T_{10} = 2 + 9 \times 3\] \[T_{10} = 2 + 27 = 29\]
29 pots are placed in the 10th row.
ii. Difference in the number of pots placed in the 5th row and the 2nd row:
Calculation: \[T_5 - T_2 = [a + (5 - 1)d] - [a + (2 - 1)d]\]
\[T_5 - T_2 = (a + 4d) - (a + d) = 3d\]
\[T_5 - T_2 = 3 \times 3 = 9\]
The difference is 9 pots.
iii. Total number of rows formed if Sanjana places 100 pots:
Given: Total sum \[S_n = 100\]
Formula: \[S_n = \frac{n}{2}[2a + (n - 1)d]\]
Substitution and Calculation: \[100 = \frac{n}{2}[2(2) + (n - 1)3]\]
\[200 = n[4 + 3n - 3]\]
\[200 = n(3n + 1)\]
\[3n^2 + n - 200 = 0\]
Factoring by splitting the middle term: \[3n^2 + 25n - 24n - 200 = 0\]
\[n(3n + 25) - 8(3n + 25) = 0\]
\[(3n + 25)(n - 8) = 0\]
\[n = -\frac{25}{3} \quad \text{or} \quad n = 8\]
Since the number of rows (n) must be a positive integer, we discard \(n = -\frac{25}{3}\).
Total number of rows formed is 8.
iv. Total number of pots placed in 12 rows:
Given: \[n = 12\]
Formula: \[S_n = \frac{n}{2}[2a + (n - 1)d]\]
Substitution and Calculation: \[S_{12} = \frac{12}{2}[2(2) + (12 - 1)3]\]
\[S_{12} = 6 \times [4 + 11 \times 3]\]
\[S_{12} = 6 \times (4 + 33)\]
\[S_{12} = 6 \times 37 = 222\]
The total number of pots placed is 222 pots.
