Find the values of k, if the points A (k+1,2k) ,B (3k,2k+3) and C (5k-1,5k) are collinear.

Explanation

To find the values of \( k \) for which the points \( A(k+1, 2k) \), \( B(3k, 2k+3) \), and \( C(5k-1, 5k) \) are collinear, we can use the concept that the area of the triangle formed by these three points must be zero.

### Step 1: Set up the area formula
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Since we want the points to be collinear, we set the area to zero:
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