cIf `a_1,a_2,a_3,..,a_n in R` then `(x-a_1)^2+(x-a_2)^2+....+(x-a_n)^2` assumes its least value at x=

Options

A. `a_(1) + a_(2) +....+a_(n)`
B. `2(a_(1) + a_(2), a_(3) +....+a_(n))`
C. `n(a_(1)+a_(2)+....+a_(n))`
D. none of these

Explanation

To find the value of \( x \) at which the expression

\[ (x - a_1)^2 + (x - a_2)^2 + \ldots + (x - a_n)^2 \]

assumes its least value, we can follow these steps:

### Step 1: Expand the Expression

We start by expanding each term in the sum:

\[ (x - a_i)^2 = x^2 - 2a_ix + a_i^2 \]

So, the entire expression becomes:

\[ \sum_{i=1}^{n} (x - a_i)^2 = \sum_{i=1}^{n} (x^2 - 2a_ix + a_i^2) \]

### Step 2: Combine Like Terms

Now, we can combine the terms:

\[ = \sum_{i=1}^{n} x^2 - \sum_{i=1}^{n} 2a_ix + \sum_{i=1}^{n} a_i^2 \]

This simplifies to:

\[ = n x^2 - 2x \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} a_i^2 \]

### Step 3: Rewrite the Expression

We can rewrite the expression as:

\[ y = n x^2 - 2 \left( \sum_{i=1}^{n} a_i \right) x + \sum_{i=1}^{n} a_i^2 \]

### Step 4: Identify the Coefficients

In this quadratic expression \( y = ax^2 + bx + c \), we identify:

- \( a = n \)
- \( b = -2 \sum_{i=1}^{n} a_i \)
- \( c = \sum_{i=1}^{n} a_i^2 \)

### Step 5: Find the Vertex

The minimum value of a quadratic function occurs at the vertex, which can be found using the formula:

\[ x = -\frac{b}{2a} \]

Substituting our values of \( a \) and \( b \):

\[ x = -\frac{-2 \sum_{i=1}^{n} a_i}{2n} = \frac{\sum_{i=1}^{n} a_i}{n} \]

### Conclusion

Thus, the value of \( x \) at which the expression assumes its least value is:

\[ x = \frac{a_1 + a_2 + \ldots + a_n}{n} \]