Let `a _(1), a_(2), a_(3)…….` be a sequence of non-zero rela numbers with are in A.P. for `k in N.` Let `f _(k)(x) =a_(k ) x ^(2) + 2a _(k+1) x + a _(k+2)`

Options

A. `f _(k) (x) =0` has real roots for each `k in N.`
B. Each of `f _(k)(x) =0` has one root in common.
C. Non-common roots of `f _(1) (x) =0 , f _(2)(x)=0, f _(3)(x)=0,……` from an A.P.
D. None of these

Answer

To solve the problem step by step, we need to analyze the function \( f_k(x) \) given in the question and check the validity of the options provided.

### Step 1: Understanding the Sequence
The sequence \( a_1, a_2, a_3, \ldots \) is in Arithmetic Progression (A.P.). This means there exists a common difference \( d \) such that:
- \( a_2 = a_1 + d \)
- \( a_3 = a_1 + 2d \)

### Step 2: Define the Function
The function is defined as:
\[ f_k(x) = a_k x^2 + 2a_{k+1} x + a_{k+2} \]
For \( k = 1 \):
\[ f_1(x) = a_1 x^2 + 2a_2 x + a_3 \]
For \( k = 2 \):
\[ f_2(x) = a_2 x^2 + 2a_3 x + a_4 \]
For \( k = 3 \):
\[ f_3(x) = a_3 x^2 + 2a_4 x + a_5 \]

### Step 3: Check Option 1
We need to check if \( f_k(x) = 0 \) has real roots for \( k \in \mathbb{N} \).
Using the quadratic formula, the roots of \( f_k(x) = 0 \) are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = a_k \), \( b = 2a_{k+1} \), and \( c = a_{k+2} \).

### Step 4: Calculate the Discriminant
The discriminant \( D \) is:
\[ D = (2a_{k+1})^2 - 4a_k a_{k+2} = 4a_{k+1}^2 - 4a_k a_{k+2} \]
For real roots, we need \( D \geq 0 \):
\[ a_{k+1}^2 \geq a_k a_{k+2} \]

### Step 5: Check Option 2
We need to check if \( f_k(x) = 0 \) has one root in common for \( k = 1 \) and \( k = 2 \).
From the previous calculations, we can analyze the roots of \( f_1(x) \) and \( f_2(x) \) to see if they share a common root.

### Step 6: Check Option 3
We need to check if the non-common roots of \( f_1(x) \), \( f_2(x) \), and \( f_3(x) \) form an A.P.
This involves finding the roots of each function and checking the condition for A.P.

### Step 7: Conclusion
After evaluating the options:
- Option 1 is correct as \( f_k(x) = 0 \) has real roots.
- Option 2 is also correct as \( f_1(x) \) and \( f_2(x) \) share a common root.
- Option 3 is incorrect as the non-common roots do not form an A.P.

### Final Answer
The correct options are 1 and 2.