The Arithmetic Mean of 13 observations is 14. If the Mean of 1st seven observations is 12 and that of last seven observations is 16, the 7th observation is-

Options

A. 12
B. 13
C. 24
D. 14

Answer

To find the 7th observation given the conditions of the problem, we can follow these steps:

### Step 1: Calculate the sum of all 13 observations
The arithmetic mean (AM) of 13 observations is given as 14. We can use the formula for arithmetic mean:

\[ \text{AM} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Thus, the sum of the 13 observations can be calculated as:

\[ \text{Sum of 13 observations} = \text{AM} \times \text{Number of observations} = 14 \times 13 = 182 \]

### Step 2: Calculate the sum of the first 7 observations
The mean of the first 7 observations is given as 12. Using the same formula:

\[ \text{Sum of first 7 observations} = 12 \times 7 = 84 \]

### Step 3: Calculate the sum of the last 7 observations
The mean of the last 7 observations is given as 16. Therefore, we can calculate:

\[ \text{Sum of last 7 observations} = 16 \times 7 = 112 \]

### Step 4: Set up the equation
We know that the last 7 observations include the 7th observation. Therefore, we can express the sum of the last 7 observations as:

\[ \text{Sum of last 7 observations} = a_7 + \text{Sum of the first 6 observations} \]

Let’s denote the sum of the first 6 observations as \(S_6\). Thus, we can write:

\[ 112 = a_7 + S_6 \]

### Step 5: Relate the sums
From Step 2, we know that:

\[ S_6 = 84 - a_7 \]

Now, substituting \(S_6\) into the equation from Step 4:

\[ 112 = a_7 + (84 - a_7) \]

### Step 6: Solve for \(a_7\)
This simplifies to:

\[ 112 = 84 \]

This equation does not directly help us find \(a_7\). Instead, we can use the total sum of all observations:

From Step 1, we have:

\[ 182 = S_6 + a_7 + (S_7 - a_7) \]

Where \(S_7\) is the sum of the last 7 observations. Since \(S_7 = 112\):

\[ 182 = 84 + a_7 + 112 - a_7 \]

This simplifies to:

\[ 182 = 196 - a_7 \]

### Step 7: Isolate \(a_7\)
Rearranging gives us:

\[ a_7 = 196 - 182 = 14 \]

### Conclusion
Thus, the 7th observation \(a_7\) is:

\[ \boxed{14} \]