There are n sets of observation given as `(1),(2, 3), (4, 5, 6), (7, 8, 9, 10),…..` The mean of the `13^("th")` set of observation is equal to

Options

A. 70
B. 80
C. 75
D. 85

Answer

To find the mean of the 13th set of observations given in the sequence `(1), (2, 3), (4, 5, 6), (7, 8, 9, 10), ...`, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Pattern of Sets**:
Each set has an increasing number of elements:
- The 1st set has 1 element: `(1)`
- The 2nd set has 2 elements: `(2, 3)`
- The 3rd set has 3 elements: `(4, 5, 6)`
- The 4th set has 4 elements: `(7, 8, 9, 10)`
- The nth set has n elements.

2. **Find the First Element of the nth Set**:
The first element of the nth set can be found by summing the number of elements in all previous sets. The total number of elements in the first (n-1) sets is:
\[ 1 + 2 + 3 + ... + (n-1) = \frac{(n-1)n}{2} \]
Therefore, the first element of the nth set is:
\[ \text{First element} = 1 + \frac{(n-1)n}{2} = \frac{n^2 - n + 2}{2} \]

3. **Find the Last Element of the nth Set**:
The last element of the nth set is the first element plus (n-1):
\[ \text{Last element} = \frac{n^2 - n + 2}{2} + (n - 1) = \frac{n^2 - n + 2 + 2n - 2}{2} = \frac{n^2 + n}{2} \]

4. **Calculate the Sum of the nth Set**:
The sum of the elements in the nth set can be calculated using the formula for the sum of an arithmetic series:
\[ S_n = \frac{n}{2} \times (\text{First element} + \text{Last element}) = \frac{n}{2} \times \left(\frac{n^2 - n + 2}{2} + \frac{n^2 + n}{2}\right) \]
Simplifying this gives:
\[ S_n = \frac{n}{2} \times \left(\frac{2n^2 + 2}{2}\right) = \frac{n(n^2 + 1)}{2} \]

5. **Calculate the Mean of the nth Set**:
The mean of the nth set is the sum of the elements divided by the number of elements:
\[ \text{Mean} = \frac{S_n}{n} = \frac{\frac{n(n^2 + 1)}{2}}{n} = \frac{n^2 + 1}{2} \]

6. **Substitute n = 13**:
Now, substituting n = 13 into the mean formula:
\[ \text{Mean} = \frac{13^2 + 1}{2} = \frac{169 + 1}{2} = \frac{170}{2} = 85 \]

### Final Answer:
The mean of the 13th set of observations is **85**.