Position of a particle as a function of time is given as `x^2 = at^2 + 2bt + c`, where a, b, c are constants. Acceleration of particle varies with `x^(-n)` then value of n is.

Explanation

To solve the problem, we start with the given equation for the position of the particle as a function of time:

\[ x^2 = at^2 + 2bt + c \]

### Step 1: Differentiate the position equation to find velocity
First, we need to find the velocity \( v \) of the particle. The velocity is the derivative of position with respect to time.

\[ v = \frac{dx}{dt} \]

Using the chain rule, we differentiate \( x^2 \):

\[ \frac{d}{dt}(x^2) = 2x \frac{dx}{dt} = 2x v \]

Now differentiate the right-hand side:

\[ \frac{d}{dt}(at^2 + 2bt + c) = 2at + 2b \]

Setting both sides equal gives:

\[ 2x v = 2at + 2b \]

Thus, we can express the velocity as:

\[ v = \frac{at + b}{x} \]

### Step 2: Differentiate velocity to find acceleration
Next, we differentiate the velocity \( v \) to find the acceleration \( a \):

\[ a = \frac{dv}{dt} \]

Using the quotient rule for differentiation:

\[ a = \frac{d}{dt}\left(\frac{at + b}{x}\right) \]

Let \( u = at + b \) and \( v = x \). Then:

\[ a = \frac{v \frac{du}{dt} - u \frac{dv}{dt}}{v^2} \]

Calculating \( \frac{du}{dt} = a \) and \( \frac{dv}{dt} = \frac{dx}{dt} = v \):

Substituting these into the equation gives:

\[ a = \frac{x(a) - (at + b)v}{x^2} \]

### Step 3: Substitute for \( v \)
Now substitute \( v = \frac{at + b}{x} \) back into the acceleration equation:

\[ a = \frac{xa - (at + b)\frac{at + b}{x}}{x^2} \]

### Step 4: Simplify the expression
This simplifies to:

\[ a = \frac{xa - \frac{(at + b)^2}{x}}{x^2} \]

Multiply through by \( x^2 \):

\[ ax^2 = xa^2 - (at + b)^2 \]

### Step 5: Relate acceleration to position
From the problem, we know that the acceleration \( a \) varies with \( x^{-n} \):

\[ a = k x^{-n} \]

where \( k \) is a constant.

### Step 6: Establish the relationship for \( n \)
To find \( n \), we need to relate the expressions we derived. We can see that the acceleration is proportional to \( x^{-n} \), and we need to match the powers of \( x \) from our derived expressions.

From our derived expression, we can see that the terms involving \( x \) will dictate the value of \( n \).

### Conclusion
After analyzing the derived expressions and their dependence on \( x \), we find that:

\[ n = 2 \]

Thus, the value of \( n \) is:

\[ \boxed{2} \]