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 pattern `(1), (2, 3), (4, 5, 6), (7, 8, 9, 10), ...`, we can follow these steps:

### Step 1: Identify the first term of the nth set
The first term of the nth set can be derived from the formula:
\[ F_n = 1 + \frac{(n-1) \cdot n}{2} \]
This formula accounts for the sum of the first \(n-1\) natural numbers, which gives us the starting point for each set.

### Step 2: Calculate the first term of the 13th set
For \(n = 13\):
\[ F_{13} = 1 + \frac{(13-1) \cdot 13}{2} \]
Calculating this:
\[ F_{13} = 1 + \frac{12 \cdot 13}{2} = 1 + \frac{156}{2} = 1 + 78 = 79 \]
Thus, the first term of the 13th set is 79.

### Step 3: Determine the number of terms in the 13th set
The number of terms in the nth set is equal to \(n\). Therefore, the 13th set has 13 terms.

### Step 4: List the terms in the 13th set
The terms in the 13th set will be:
\[ 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91 \]

### Step 5: Calculate the sum of the terms in the 13th set
The sum of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]
Where:
- \(n\) is the number of terms (13),
- \(a\) is the first term (79),
- \(d\) is the common difference (1).

Substituting the values:
\[ S_{13} = \frac{13}{2} \cdot (2 \cdot 79 + (13-1) \cdot 1) \]
Calculating this:
\[ S_{13} = \frac{13}{2} \cdot (158 + 12) = \frac{13}{2} \cdot 170 = 13 \cdot 85 = 1105 \]

### Step 6: Calculate the mean of the 13th set
The mean is given by:
\[ \text{Mean} = \frac{S_n}{n} = \frac{1105}{13} = 85 \]

Thus, the mean of the 13th set of observations is **85**.

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