The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), …..and so on. Then, the sum of the numbers in the 15th group is equal to
- 6119
- 7471
- 4941
- 6090
The Correct Option is A
Solution and Explanation
The natural numbers are organized into sets like (1), (2, 3, 4), (5, 6, 7, 8, 9), and this pattern continues indefinitely.
Notice that the first set concludes at \( 1^2 \), the second set concludes at \( 2^2 \), and the third set concludes at \( 3^2 \).
Step 1: Determine the 15th Set
The 15th set concludes with \( 15^2 \), which equals 225. The 14th set concludes with \( 14^2 \), which equals 196. Therefore, the 15th set consists of the numbers from 197 to 225, inclusive.
Step 2: Calculate the Total of the Numbers in the 15th Set
The total of the numbers in the 15th set can be calculated using the formula for the sum of an arithmetic series: \[ \text{Sum} = \frac{n}{2} \times (\text{First term} + \text{Last term}) \] Where: - The first term is 197, - The last term is 225, - The number of terms, \( n \), is \( 225 - 197 + 1 = 29 \). Substituting into the formula: \[ \text{Sum} = \frac{29}{2} \times (197 + 225) = \frac{29}{2} \times 422 = 6119 \]
Final Answer:
The total of the numbers in the 15th set is \( \boxed{6119} \).
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