The mean of 3, a+2, 8, 12, 2a-1 and 6 is 7, find the value of a.

Answer

To find the value of \( a \) given that the mean of the numbers \( 3, a + 2, 8, 12, 2a - 1, \) and \( 6 \) is \( 7 \), we can follow these steps:

### Step 1: Set up the equation for the mean
The mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers. In this case, we have 6 numbers, and we know the mean is \( 7 \).

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

Given that the mean is \( 7 \) and there are \( 6 \) observations, we can write:

\[ 7 = \frac{3 + (a + 2) + 8 + 12 + (2a - 1) + 6}{6} \]

### Step 2: Simplify the equation
Now, we will simplify the numerator:

\[ 3 + (a + 2) + 8 + 12 + (2a - 1) + 6 \]

Combining like terms:

\[ = 3 + 2 + 8 + 12 - 1 + 6 + a + 2a \]
\[ = (3 + 2 + 8 + 12 - 1 + 6) + (a + 2a) \]
\[ = 30 + 3a \]

### Step 3: Substitute back into the mean equation
Now substitute this back into the mean equation:

\[ 7 = \frac{30 + 3a}{6} \]

### Step 4: Eliminate the fraction
To eliminate the fraction, multiply both sides by \( 6 \):

\[ 7 \times 6 = 30 + 3a \]
\[ 42 = 30 + 3a \]

### Step 5: Solve for \( a \)
Now, isolate \( 3a \):

\[ 3a = 42 - 30 \]
\[ 3a = 12 \]

Now, divide by \( 3 \):

\[ a = \frac{12}{3} \]
\[ a = 4 \]

### Conclusion
The value of \( a \) is \( 4 \).

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