If the mean of a, b, c, d and e is 28, mean of a, c and e is 24 and mean of b and d is `n^2` - 2, then the value of n is ± k, where k is

Answer

To solve the problem step by step, we will use the information given about the means of the sets of variables \(a\), \(b\), \(c\), \(d\), and \(e\).

### Step 1: Calculate the sum of \(a\), \(b\), \(c\), \(d\), and \(e\)
The mean of \(a\), \(b\), \(c\), \(d\), and \(e\) is given as 28.

Using the formula for mean:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
We can write:
\[ \frac{a + b + c + d + e}{5} = 28 \]
Multiplying both sides by 5 gives:
\[ a + b + c + d + e = 28 \times 5 = 140 \]

### Step 2: Calculate the sum of \(a\), \(c\), and \(e\)
The mean of \(a\), \(c\), and \(e\) is given as 24.

Using the same formula for mean:
\[ \frac{a + c + e}{3} = 24 \]
Multiplying both sides by 3 gives:
\[ a + c + e = 24 \times 3 = 72 \]

### Step 3: Calculate the sum of \(b\) and \(d\)
From the previous steps, we have:
\[ a + b + c + d + e = 140 \]
Substituting \(a + c + e = 72\) into this equation:
\[ 72 + b + d = 140 \]
Now, solving for \(b + d\):
\[ b + d = 140 - 72 = 68 \]

### Step 4: Calculate the mean of \(b\) and \(d\)
The mean of \(b\) and \(d\) can be calculated as:
\[ \frac{b + d}{2} = \frac{68}{2} = 34 \]

### Step 5: Set up the equation for \(n\)
We are given that the mean of \(b\) and \(d\) is also equal to \(n^2 - 2\):
\[ n^2 - 2 = 34 \]
Now, solving for \(n^2\):
\[ n^2 = 34 + 2 = 36 \]

### Step 6: Solve for \(n\)
Taking the square root of both sides gives:
\[ n = \pm 6 \]

### Step 7: Identify \(k\)
The problem states that \(n\) is \( \pm k\), where \(k\) is the value we need to find. Thus, we have:
\[ k = 6 \]

### Final Answer
The value of \(k\) is \(6\).

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