If `sum_(i=1)^(n)(x_(i)-2)=110, sum_(i=1)^(n)(x_(i)-5)=20`, then what is the mean?

Options

A. `11//2`
B. `2//11`
C. `17//3`
D. `17//9`

Explanation

To solve the problem, we need to determine the mean of the series given the two summation equations. Let's break down the solution step by step:

### Step 1: Interpret the Given Summations
We are given two equations:
1. \(\sum_{i=1}^{n}(x_{i} - 2) = 110\)
2. \(\sum_{i=1}^{n}(x_{i} - 5) = 20\)

### Step 2: Rewrite the Summations
From the first equation:
\[ \sum_{i=1}^{n} x_{i} - 2n = 110 \]
This implies:
\[ \sum_{i=1}^{n} x_{i} = 110 + 2n \quad \text{(Equation 1)} \]

From the second equation:
\[ \sum_{i=1}^{n} x_{i} - 5n = 20 \]
This implies:
\[ \sum_{i=1}^{n} x_{i} = 20 + 5n \quad \text{(Equation 2)} \]

### Step 3: Set the Two Equations Equal
Since both expressions equal \(\sum_{i=1}^{n} x_{i}\), we can set Equation 1 equal to Equation 2:
\[ 110 + 2n = 20 + 5n \]

### Step 4: Solve for \(n\)
Rearranging the equation:
\[ 110 - 20 = 5n - 2n \]
\[ 90 = 3n \]
\[ n = \frac{90}{3} = 30 \]

### Step 5: Substitute \(n\) Back to Find the Sum
Now that we have \(n\), we can substitute it back into either Equation 1 or Equation 2 to find \(\sum_{i=1}^{n} x_{i}\). Using Equation 1:
\[ \sum_{i=1}^{n} x_{i} = 110 + 2(30) \]
\[ \sum_{i=1}^{n} x_{i} = 110 + 60 = 170 \]

### Step 6: Calculate the Mean
The mean is calculated as:
\[ \text{Mean} = \frac{\sum_{i=1}^{n} x_{i}}{n} \]
Substituting the values we found:
\[ \text{Mean} = \frac{170}{30} = \frac{17}{3} \]

### Final Answer
The mean of the series is \(\frac{17}{3}\).

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